VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.0%

Time: 13.1s

Alternatives: 23

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot t\_0, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -1.2e+167)
     (/ (- -1.0 (* (cos B) x)) (sin B))
     (if (<= F 1e+147)
       (fma F (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) t_0) (/ (- x) (tan B)))
       (- t_0 (* (/ x (sin B)) (cos B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -1.2e+167) {
		tmp = (-1.0 - (cos(B) * x)) / sin(B);
	} else if (F <= 1e+147) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * t_0), (-x / tan(B)));
	} else {
		tmp = t_0 - ((x / sin(B)) * cos(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.2e+167)
		tmp = Float64(Float64(-1.0 - Float64(cos(B) * x)) / sin(B));
	elseif (F <= 1e+147)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * t_0), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(t_0 - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e+167], N[(N[(-1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+147], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * t$95$0), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999999e167

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1 - \cos B \cdot x}{\color{blue}{\sin B}} \]

   if -1.19999999999999999e167 < F < 9.9999999999999998e146

   1. Initial program 94.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   if 9.9999999999999998e146 < F

   1. Initial program 24.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x \cdot \cos B}{\sin B} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} - \frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\sin B} - \frac{\color{blue}{\cos B \cdot x}}{\sin B} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\cos B} \cdot \frac{x}{\sin B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sin B} - \cos B \cdot \color{blue}{\frac{x}{\sin B}} \]

      9. lower-sin.f64 99.9

         \[\leadsto \frac{1}{\sin B} - \cos B \cdot \frac{x}{\color{blue}{\sin B}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -320000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -320000000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 110000000.0)
       (fma (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) F t_0)
       (fma (/ 1.0 (sin B)) 1.0 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -320000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 110000000.0) {
		tmp = fma((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), F, t_0);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -320000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 110000000.0)
		tmp = fma(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), F, t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -320000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 110000000.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F + t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.2e8

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   if -3.2e8 < F < 1.1e8

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \cdot F + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}, F, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   if 1.1e8 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 105000000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.2e+167)
     (/ (- -1.0 (* (cos B) x)) (sin B))
     (if (<= F 105000000.0)
       (fma t_0 (/ (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ 1.0 F)) t_1)
       (fma t_0 1.0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -1.2e+167) {
		tmp = (-1.0 - (cos(B) * x)) / sin(B);
	} else if (F <= 105000000.0) {
		tmp = fma(t_0, (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) / (1.0 / F)), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.2e+167)
		tmp = Float64(Float64(-1.0 - Float64(cos(B) * x)) / sin(B));
	elseif (F <= 105000000.0)
		tmp = fma(t_0, Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) / Float64(1.0 / F)), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e+167], N[(N[(-1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 105000000.0], N[(t$95$0 * N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 / F), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999999e167

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1 - \cos B \cdot x}{\color{blue}{\sin B}} \]

   if -1.19999999999999999e167 < F < 1.05e8

   1. Initial program 94.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   if 1.05e8 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 105000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+22)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 105000000.0)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (* (cos B) x)) (sin B))
     (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+22) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 105000000.0) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (cos(B) * x)) / sin(B);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 105000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 105000000.0], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e22

   1. Initial program 53.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   if -2e22 < F < 1.05e8

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   if 1.05e8 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.4)
     (/ (- (/ F (sqrt (fma x 2.0 2.0))) (* (cos B) x)) (sin B))
     (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.4) {
		tmp = ((F / sqrt(fma(x, 2.0, 2.0))) - (cos(B) * x)) / sin(B);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, 2.0))) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{2 + 2 \cdot x}}} - \cos B \cdot x}{\sin B} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{2 + 2 \cdot x}}} - \cos B \cdot x}{\sin B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{F}{\sqrt{\color{blue}{2 \cdot x + 2}}} - \cos B \cdot x}{\sin B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{F}{\sqrt{\color{blue}{x \cdot 2} + 2}} - \cos B \cdot x}{\sin B} \]

      4. lower-fma.f64 98.0

         \[\leadsto \frac{\frac{F}{\sqrt{\color{blue}{\mathsf{fma}\left(x, 2, 2\right)}}} - \cos B \cdot x}{\sin B} \]

   9. Applied rewrites 98.0%

      \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{\mathsf{fma}\left(x, 2, 2\right)}}} - \cos B \cdot x}{\sin B} \]

   if 1.3999999999999999 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1550:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2500.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1550.0)
     (+
      (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) (/ F B))
      (* (/ -1.0 (tan B)) x))
     (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2500.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1550.0) {
		tmp = (sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * (F / B)) + ((-1.0 / tan(B)) * x);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2500.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1550.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * Float64(F / B)) + Float64(Float64(-1.0 / tan(B)) * x));
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2500.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1550.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2500

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   if -2500 < F < 1550

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \]

      10. lower-fma.f64 84.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]

   5. Applied rewrites 84.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1550 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) + \frac{\mathsf{neg}\left(x\right)}{\tan B}} \]

      2. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(F \cdot \color{blue}{\frac{1}{\sin B}}\right) \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      8. div-inv N/A

         \[\leadsto \frac{1 \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B \cdot \frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      10. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}} + \frac{\mathsf{neg}\left(x\right)}{\tan B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{1}{F}}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right)} \]

   6. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\frac{1}{F}}, \frac{-x}{\tan B}\right)} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1550:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1820:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2500.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1820.0)
     (+
      (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) (/ F B))
      (* (/ -1.0 (tan B)) x))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2500.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1820.0) {
		tmp = (sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * (F / B)) + ((-1.0 / tan(B)) * x);
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2500.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1820.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * Float64(F / B)) + Float64(Float64(-1.0 / tan(B)) * x));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2500.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1820.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2500

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   if -2500 < F < 1820

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \]

      10. lower-fma.f64 84.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]

   5. Applied rewrites 84.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1820 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1820:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1820:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2500.0)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 1820.0)
       (+
        (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) (/ F B))
        (* (/ -1.0 (tan B)) x))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -2500.0) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 1820.0) {
		tmp = (sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * (F / B)) + ((-1.0 / tan(B)) * x);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2500.0)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 1820.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * Float64(F / B)) + Float64(Float64(-1.0 / tan(B)) * x));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2500.0], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1820.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2500

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \frac{-1 - \cos B \cdot x}{\color{blue}{\sin B}} \]

   if -2500 < F < 1820

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \]

      10. lower-fma.f64 84.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]

   5. Applied rewrites 84.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1820 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1820:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2500.0)
   (/ (- -1.0 (* (cos B) x)) (sin B))
   (if (<= F 5.3e-40)
     (+
      (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) (/ F B))
      (* (/ -1.0 (tan B)) x))
     (if (<= F 1.35e+154)
       (/ (fma (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) F (- x)) (sin B))
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2500.0) {
		tmp = (-1.0 - (cos(B) * x)) / sin(B);
	} else if (F <= 5.3e-40) {
		tmp = (sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * (F / B)) + ((-1.0 / tan(B)) * x);
	} else if (F <= 1.35e+154) {
		tmp = fma(sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, -x) / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2500.0)
		tmp = Float64(Float64(-1.0 - Float64(cos(B) * x)) / sin(B));
	elseif (F <= 5.3e-40)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * Float64(F / B)) + Float64(Float64(-1.0 / tan(B)) * x));
	elseif (F <= 1.35e+154)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2500.0], N[(N[(-1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.3e-40], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e+154], N[(N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2500

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \frac{-1 - \cos B \cdot x}{\color{blue}{\sin B}} \]

   if -2500 < F < 5.3000000000000002e-40

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \]

      10. lower-fma.f64 84.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]

   5. Applied rewrites 84.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 5.3000000000000002e-40 < F < 1.35000000000000003e154

   1. Initial program 93.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{\sin B} \]
   8. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{\sin B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      14. lower-neg.f64 91.9

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, \color{blue}{-x}\right)}{\sin B} \]

   9. Applied rewrites 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}}{\sin B} \]

   if 1.35000000000000003e154 < F

   1. Initial program 22.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 30.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.8%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2500:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \frac{F}{B} + \frac{-1}{\tan B} \cdot x\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -3.15e-16)
   (/ (* (- x) (cos B)) (sin B))
   (if (<= x 7.2e-58)
     (/ (fma (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) F (- x)) (sin B))
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -3.15e-16) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (x <= 7.2e-58) {
		tmp = fma(sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, -x) / sin(B);
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -3.15e-16)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (x <= 7.2e-58)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -3.15e-16], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-58], N[(N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1499999999999999e-16

   1. Initial program 52.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{\cos B}{\sin B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\color{blue}{\cos B}}{\sin B} \]

      8. lower-sin.f64 93.4

         \[\leadsto \left(-x\right) \cdot \frac{\cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.6%

      \[\leadsto \frac{\cos B \cdot \left(-x\right)}{\color{blue}{\sin B}} \]

   if -3.1499999999999999e-16 < x < 7.20000000000000019e-58

   1. Initial program 74.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{\sin B} \]
   8. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{\sin B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      14. lower-neg.f64 69.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, \color{blue}{-x}\right)}{\sin B} \]

   9. Applied rewrites 69.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}}{\sin B} \]

   if 7.20000000000000019e-58 < x

   1. Initial program 85.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{\cos B}{\sin B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\color{blue}{\cos B}}{\sin B} \]

      8. lower-sin.f64 92.0

         \[\leadsto \left(-x\right) \cdot \frac{\cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -3.15 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -3.15e-16)
     t_0
     (if (<= x 7.2e-58)
       (/ (fma (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) F (- x)) (sin B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -3.15e-16) {
		tmp = t_0;
	} else if (x <= 7.2e-58) {
		tmp = fma(sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, -x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -3.15e-16)
		tmp = t_0;
	elseif (x <= 7.2e-58)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), F, Float64(-x)) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.15e-16], t$95$0, If[LessEqual[x, 7.2e-58], N[(N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1499999999999999e-16 or 7.20000000000000019e-58 < x

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{\cos B}{\sin B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\color{blue}{\cos B}}{\sin B} \]

      8. lower-sin.f64 92.4

         \[\leadsto \left(-x\right) \cdot \frac{\cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if -3.1499999999999999e-16 < x < 7.20000000000000019e-58

   1. Initial program 74.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{\sin B} \]
   8. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{\sin B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{\sin B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{\sin B} \]

      14. lower-neg.f64 69.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, \color{blue}{-x}\right)}{\sin B} \]

   9. Applied rewrites 69.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, -x\right)}}{\sin B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.4e-9)
   (/ (fma (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) F (- x)) B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.4e-9) {
		tmp = fma(sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))), F, -x) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.4e-9)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.4e-9], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3999999999999997e-9

   1. Initial program 73.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 65.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   if 4.3999999999999997e-9 < B

   1. Initial program 82.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{\cos B}{\sin B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\color{blue}{\cos B}}{\sin B} \]

      8. lower-sin.f64 47.7

         \[\leadsto \left(-x\right) \cdot \frac{\cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, -0.5, 1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e+167)
   (/
    (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) (- -1.0 x))
    B)
   (if (<= F 1500000.0)
     (/ (fma (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) F (- x)) B)
     (/ (fma (/ (fma x 2.0 2.0) (* F F)) -0.5 (- 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e+167) {
		tmp = fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), (-1.0 - x)) / B;
	} else if (F <= 1500000.0) {
		tmp = fma(sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))), F, -x) / B;
	} else {
		tmp = fma((fma(x, 2.0, 2.0) / (F * F)), -0.5, (1.0 - x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e+167)
		tmp = Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), Float64(-1.0 - x)) / B);
	elseif (F <= 1500000.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))), F, Float64(-x)) / B);
	else
		tmp = Float64(fma(Float64(fma(x, 2.0, 2.0) / Float64(F * F)), -0.5, Float64(1.0 - x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e+167], N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1500000.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999999e167

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{\left(-1 \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) - \frac{1}{6}\right)\right) - 1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{\color{blue}{B}} \]

   if -1.19999999999999999e167 < F < 1.5e6

   1. Initial program 94.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 56.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   if 1.5e6 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 42.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, -0.5, 1 - x\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-113}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 3700000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e-29)
   (/ (fma (* (/ x (* F F)) 2.0) 0.5 (- -1.0 x)) B)
   (if (<= F 1.22e-113)
     (/ (- x) B)
     (if (<= F 3700000.0)
       (/ (* (sqrt (/ 1.0 (fma F F 2.0))) F) B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-29) {
		tmp = fma(((x / (F * F)) * 2.0), 0.5, (-1.0 - x)) / B;
	} else if (F <= 1.22e-113) {
		tmp = -x / B;
	} else if (F <= 3700000.0) {
		tmp = (sqrt((1.0 / fma(F, F, 2.0))) * F) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-29)
		tmp = Float64(fma(Float64(Float64(x / Float64(F * F)) * 2.0), 0.5, Float64(-1.0 - x)) / B);
	elseif (F <= 1.22e-113)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 3700000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * F) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-29], N[(N[(N[(N[(x / N[(F * F), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.22e-113], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 3700000.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.99999999999999943e-30

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 54.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, 0.5, -1 - x\right)}{B} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(2 \cdot \frac{x}{{F}^{2}}, \frac{1}{2}, -1 - x\right)}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.3%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B} \]

   if -9.99999999999999943e-30 < F < 1.21999999999999995e-113

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.7%

      \[\leadsto \frac{-x}{B} \]

   if 1.21999999999999995e-113 < F < 3.7e6

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 68.3

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.3%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]

   if 3.7e6 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 43.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 44.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-113}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 3700000:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e-29)
   (/ (fma (* (/ x (* F F)) 2.0) 0.5 (- -1.0 x)) B)
   (if (<= F 1.22e-113)
     (/ (- x) B)
     (if (<= F 3700000.0)
       (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F B))
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-29) {
		tmp = fma(((x / (F * F)) * 2.0), 0.5, (-1.0 - x)) / B;
	} else if (F <= 1.22e-113) {
		tmp = -x / B;
	} else if (F <= 3700000.0) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-29)
		tmp = Float64(fma(Float64(Float64(x / Float64(F * F)) * 2.0), 0.5, Float64(-1.0 - x)) / B);
	elseif (F <= 1.22e-113)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 3700000.0)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-29], N[(N[(N[(N[(x / N[(F * F), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.22e-113], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 3700000.0], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.99999999999999943e-30

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 54.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, 0.5, -1 - x\right)}{B} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(2 \cdot \frac{x}{{F}^{2}}, \frac{1}{2}, -1 - x\right)}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.3%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B} \]

   if -9.99999999999999943e-30 < F < 1.21999999999999995e-113

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.7%

      \[\leadsto \frac{-x}{B} \]

   if 1.21999999999999995e-113 < F < 3.7e6

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 68.3

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   if 3.7e6 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 43.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 51.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, -0.5, 1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e+167)
   (/
    (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) (- -1.0 x))
    B)
   (if (<= F 1000000.0)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/ (fma (/ (fma x 2.0 2.0) (* F F)) -0.5 (- 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e+167) {
		tmp = fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), (-1.0 - x)) / B;
	} else if (F <= 1000000.0) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = fma((fma(x, 2.0, 2.0) / (F * F)), -0.5, (1.0 - x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e+167)
		tmp = Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), Float64(-1.0 - x)) / B);
	elseif (F <= 1000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(fma(Float64(fma(x, 2.0, 2.0) / Float64(F * F)), -0.5, Float64(1.0 - x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e+167], N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1000000.0], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999999e167

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{\left(-1 \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) - \frac{1}{6}\right)\right) - 1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{\color{blue}{B}} \]

   if -1.19999999999999999e167 < F < 1e6

   1. Initial program 94.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 56.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 1e6 < F

   1. Initial program 54.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 42.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, -0.5, 1 - x\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 51.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e+167)
   (/
    (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) (- -1.0 x))
    B)
   (if (<= F 150000000.0)
     (/ (fma (sqrt (/ 1.0 (fma F F 2.0))) F (- x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e+167) {
		tmp = fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), (-1.0 - x)) / B;
	} else if (F <= 150000000.0) {
		tmp = fma(sqrt((1.0 / fma(F, F, 2.0))), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e+167)
		tmp = Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), Float64(-1.0 - x)) / B);
	elseif (F <= 150000000.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(F, F, 2.0))), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e+167], N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 150000000.0], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999999e167

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{\left(-1 \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) - \frac{1}{6}\right)\right) - 1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{\color{blue}{B}} \]

   if -1.19999999999999999e167 < F < 1.5e8

   1. Initial program 94.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 56.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, F, -x\right)}{B} \]

   if 1.5e8 < F

   1. Initial program 54.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 43.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-117}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e-29)
   (/ (fma (* (/ x (* F F)) 2.0) 0.5 (- -1.0 x)) B)
   (if (<= F 1.9e-117) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-29) {
		tmp = fma(((x / (F * F)) * 2.0), 0.5, (-1.0 - x)) / B;
	} else if (F <= 1.9e-117) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-29)
		tmp = Float64(fma(Float64(Float64(x / Float64(F * F)) * 2.0), 0.5, Float64(-1.0 - x)) / B);
	elseif (F <= 1.9e-117)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-29], N[(N[(N[(N[(x / N[(F * F), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-117], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999943e-30

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 54.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, 2, 2\right)}{F \cdot F}, 0.5, -1 - x\right)}{B} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(2 \cdot \frac{x}{{F}^{2}}, \frac{1}{2}, -1 - x\right)}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.3%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{F \cdot F} \cdot 2, 0.5, -1 - x\right)}{B} \]

   if -9.99999999999999943e-30 < F < 1.89999999999999986e-117

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

      \[\leadsto \frac{-x}{B} \]

   if 1.89999999999999986e-117 < F

   1. Initial program 65.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 49.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-117}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e-37)
   (/
    (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) (- -1.0 x))
    B)
   (if (<= F 1.9e-117) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-37) {
		tmp = fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), (-1.0 - x)) / B;
	} else if (F <= 1.9e-117) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.6e-37)
		tmp = Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), Float64(-1.0 - x)) / B);
	elseif (F <= 1.9e-117)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.6e-37], N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-117], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.59999999999999963e-37

   1. Initial program 62.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{\cos B}{\sin B}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} + \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{\cos B}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{\frac{\cos B}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\color{blue}{\cos B}}{\sin B}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\color{blue}{\sin B}}, \mathsf{neg}\left(\frac{1}{\sin B}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \frac{\color{blue}{-1}}{\sin B}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{\cos B}{\sin B}, \color{blue}{\frac{-1}{\sin B}}\right) \]

      14. lower-sin.f64 90.0

          \[\leadsto \mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\color{blue}{\sin B}}\right) \]

   5. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{\cos B}{\sin B}, \frac{-1}{\sin B}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{\left(-1 \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) - \frac{1}{6}\right)\right) - 1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1 - x\right)}{\color{blue}{B}} \]

   if -9.59999999999999963e-37 < F < 1.89999999999999986e-117

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.8

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

      \[\leadsto \frac{-x}{B} \]

   if 1.89999999999999986e-117 < F

   1. Initial program 65.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 49.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-117}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e-37)
   (/ (- -1.0 x) B)
   (if (<= F 1.9e-117) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-37) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-117) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.6d-37)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.9d-117) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-37) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-117) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.6e-37:
		tmp = (-1.0 - x) / B
	elif F <= 1.9e-117:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.6e-37)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.9e-117)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.6e-37)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.9e-117)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.6e-37], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-117], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.59999999999999963e-37

   1. Initial program 62.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 55.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -9.59999999999999963e-37 < F < 1.89999999999999986e-117

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.8

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

      \[\leadsto \frac{-x}{B} \]

   if 1.89999999999999986e-117 < F

   1. Initial program 65.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 49.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.8% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e-37) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-37) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.6d-37)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-37) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.6e-37:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.6e-37)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.6e-37)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.6e-37], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.59999999999999963e-37

   1. Initial program 62.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 55.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -9.59999999999999963e-37 < F

   1. Initial program 80.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 49.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.1%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 29.7% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 75.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. lower-neg.f64 50.9

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

5. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 32.5%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Alternative 23: 10.2% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. lower-neg.f64 50.9

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

5. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

6. Taylor expanded in F around -inf

   \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
7. Step-by-step derivation

8. Applied rewrites 31.4%

   \[\leadsto \frac{-1 - x}{B} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{B} \]
10. Step-by-step derivation

11. Applied rewrites 8.5%

    \[\leadsto \frac{-1}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))