VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.7%

Time: 14.9s

Alternatives: 21

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: 77.1% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, -t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e+139)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.5e+21)
       (fma (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) F (- t_0))
       (fma (/ -1.0 (tan B)) x (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5e+139) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.5e+21) {
		tmp = fma((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), F, -t_0);
	} else {
		tmp = fma((-1.0 / tan(B)), x, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5e+139)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.5e+21)
		tmp = fma(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), F, Float64(-t_0));
	else
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+139], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.5e+21], N[(N[(N[Power[N[(2.0 * x + 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[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e139

   1. Initial program 42.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-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 56.0%

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

   6. Applied rewrites 56.1%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -1.5e139 < F < 1.5e21

   1. Initial program 98.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-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 99.7%

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

   if 1.5e21 < F

   1. Initial program 59.8%

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

   3. Taylor expanded in F around 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ t_2 := \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_3 := \frac{\mathsf{fma}\left(F, t\_2, -x\right)}{\sin B}\\ t_4 := t\_0 + t\_2 \cdot \frac{F}{B}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1
         (+
          t_0
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) (/ -1.0 2.0)))))
        (t_2 (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))))
        (t_3 (/ (fma F t_2 (- x)) (sin B)))
        (t_4 (+ t_0 (* t_2 (/ F B)))))
   (if (<= t_1 -200.0)
     t_4
     (if (<= t_1 -1e-154)
       t_3
       (if (<= t_1 1e-12)
         t_4
         (if (<= t_1 10.0) t_3 (if (<= t_1 1e+308) t_4 (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (2.0 * x)), (-1.0 / 2.0)));
	double t_2 = sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0))));
	double t_3 = fma(F, t_2, -x) / sin(B);
	double t_4 = t_0 + (t_2 * (F / B));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_4;
	} else if (t_1 <= -1e-154) {
		tmp = t_3;
	} else if (t_1 <= 1e-12) {
		tmp = t_4;
	} else if (t_1 <= 10.0) {
		tmp = t_3;
	} else if (t_1 <= 1e+308) {
		tmp = t_4;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))))
	t_2 = sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0))))
	t_3 = Float64(fma(F, t_2, Float64(-x)) / sin(B))
	t_4 = Float64(t_0 + Float64(t_2 * Float64(F / B)))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_4;
	elseif (t_1 <= -1e-154)
		tmp = t_3;
	elseif (t_1 <= 1e-12)
		tmp = t_4;
	elseif (t_1 <= 10.0)
		tmp = t_3;
	elseif (t_1 <= 1e+308)
		tmp = t_4;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(F * t$95$2 + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(t$95$2 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$4, If[LessEqual[t$95$1, -1e-154], t$95$3, If[LessEqual[t$95$1, 1e-12], t$95$4, If[LessEqual[t$95$1, 10.0], t$95$3, If[LessEqual[t$95$1, 1e+308], t$95$4, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -200 or -9.9999999999999997e-155 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 9.9999999999999998e-13 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1e308

   1. Initial program 87.7%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -200 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -9.9999999999999997e-155 or 9.9999999999999998e-13 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 88.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 88.0%

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

   6. Applied rewrites 87.9%

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

   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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 74.5

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

   9. Applied rewrites 74.5%

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

   if 1e308 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 20.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 80.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -200:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+75)
   (/ (fma (cos B) (- x) -1.0) (sin B))
   (if (<= F 1.5e+21)
     (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x (tan B)))
     (fma (/ -1.0 (tan B)) x (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+75) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 1.5e+21) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / tan(B));
	} else {
		tmp = fma((-1.0 / tan(B)), x, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+75)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 1.5e+21)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+75], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+21], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000002e75

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-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 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -5.0000000000000002e75 < F < 1.5e21

   1. Initial program 98.8%

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

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

   6. Applied rewrites 99.6%

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

   if 1.5e21 < F

   1. Initial program 59.8%

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

   3. Taylor expanded in F around 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.001:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.001)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 2.0)))) t_0)
       (fma (/ -1.0 (tan B)) x (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.001) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0;
	} else {
		tmp = fma((-1.0 / tan(B)), x, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.001)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0);
	else
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.001], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6499999999999999

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-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 75.2%

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -1.6499999999999999 < F < 1e-3

   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. 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%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 1e-3 < F

   1. Initial program 62.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.3%

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

Alternative 5: 91.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot t\_0 + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (tan B))))
   (if (<= F -165000000000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1.02e-5)
       (+ (* x t_0) (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
       (fma t_0 x (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / tan(B);
	double tmp;
	if (F <= -165000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.02e-5) {
		tmp = (x * t_0) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = fma(t_0, x, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / tan(B))
	tmp = 0.0
	if (F <= -165000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.02e-5)
		tmp = Float64(Float64(x * t_0) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	else
		tmp = fma(t_0, x, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -165000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-5], N[(N[(x * t$95$0), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e11

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 74.8%

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -1.65e11 < F < 1.0200000000000001e-5

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 1.0200000000000001e-5 < F

   1. Initial program 62.9%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{+111}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, t\_0, -x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t\_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0))))))
   (if (<= F -8.8e+111)
     (- (/ x (tan B)))
     (if (<= F -6e-100)
       (/ (fma F t_0 (- x)) (sin B))
       (if (<= F 1.02e-5)
         (+ (* x (/ -1.0 (tan B))) (* t_0 (/ F B)))
         (/ (- 1.0 (* x (cos B))) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0))));
	double tmp;
	if (F <= -8.8e+111) {
		tmp = -(x / tan(B));
	} else if (F <= -6e-100) {
		tmp = fma(F, t_0, -x) / sin(B);
	} else if (F <= 1.02e-5) {
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -8.8e+111)
		tmp = Float64(-Float64(x / tan(B)));
	elseif (F <= -6e-100)
		tmp = Float64(fma(F, t_0, Float64(-x)) / sin(B));
	elseif (F <= 1.02e-5)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -8.8e+111], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -6e-100], N[(N[(F * t$95$0 + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-5], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.79999999999999994e111

   1. Initial program 52.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 61.6

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

   if -8.79999999999999994e111 < F < -6.0000000000000001e-100

   1. Initial program 96.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-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 99.5%

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

   6. Applied rewrites 99.5%

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

   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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 83.9

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

   9. Applied rewrites 83.9%

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

   if -6.0000000000000001e-100 < F < 1.0200000000000001e-5

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if 1.0200000000000001e-5 < F

   1. Initial program 62.9%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

      \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{+111}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -165000000000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.02e-5)
     (+
      (* x (/ -1.0 (tan B)))
      (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -165000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.02e-5) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -165000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.02e-5)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -165000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-5], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e11

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 74.8%

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -1.65e11 < F < 1.0200000000000001e-5

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 1.0200000000000001e-5 < F

   1. Initial program 62.9%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -165000000000.0)
   (/ (fma (cos B) (- x) -1.0) (sin B))
   (if (<= F 1.02e-5)
     (+
      (* x (/ -1.0 (tan B)))
      (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -165000000000.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 1.02e-5) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -165000000000.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 1.02e-5)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -165000000000.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-5], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e11

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 74.8%

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -1.65e11 < F < 1.0200000000000001e-5

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 1.0200000000000001e-5 < F

   1. Initial program 62.9%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -165000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -0.16:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -0.16)
     t_0
     (if (<= x 2.2e-64)
       (/ (fma F (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (- x)) (sin B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -0.16) {
		tmp = t_0;
	} else if (x <= 2.2e-64) {
		tmp = fma(F, sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), -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 <= -0.16)
		tmp = t_0;
	elseif (x <= 2.2e-64)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), 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, -0.16], t$95$0, If[LessEqual[x, 2.2e-64], N[(N[(F * N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.160000000000000003 or 2.2e-64 < x

   1. Initial program 84.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 90.0

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

   5. Applied rewrites 90.0%

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

   7. Applied rewrites 90.2%

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

   if -0.160000000000000003 < x < 2.2e-64

   1. Initial program 75.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 77.1%

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

   6. Applied rewrites 77.2%

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

   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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 66.5

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

   9. Applied rewrites 66.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.16:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-112}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -2.05e-81)
     t_0
     (if (<= x 7e-112) (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -2.05e-81) {
		tmp = t_0;
	} else if (x <= 7e-112) {
		tmp = F / (sin(B) * sqrt(fma(F, F, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.05e-81)
		tmp = t_0;
	elseif (x <= 7e-112)
		tmp = Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0))));
	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, -2.05e-81], t$95$0, If[LessEqual[x, 7e-112], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.04999999999999992e-81 or 6.99999999999999988e-112 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 82.7%

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

   if -2.04999999999999992e-81 < x < 6.99999999999999988e-112

   1. Initial program 74.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 76.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \]

      8. lower-fma.f64 57.5

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 59.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-81}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-112}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.5e+161)
   (/ -1.0 (sin B))
   (if (<= F 3.6e+21)
     (/ (fma F (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (- x)) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e+161) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.6e+21) {
		tmp = fma(F, sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), -x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.5e+161)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.6e+21)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), Float64(-x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.5e+161], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e+21], N[(N[(F * N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.4999999999999995e161

   1. Initial program 40.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 52.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \]

      8. lower-fma.f64 1.9

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

   7. Applied rewrites 1.9%

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

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.7%

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

   if -7.4999999999999995e161 < F < 3.6e21

   1. Initial program 98.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if 3.6e21 < F

   1. Initial program 59.2%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.4%

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+79}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+79)
   (/ (- -1.0 x) B)
   (if (<= F 3.6e+21)
     (/ (fma F (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (- x)) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+79) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.6e+21) {
		tmp = fma(F, sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), -x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+79)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.6e+21)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), Float64(-x)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+79], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.6e+21], N[(N[(F * N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999967e78

   1. Initial program 58.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 27.0

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 38.7%

      \[\leadsto \frac{-1 - x}{B} \]

   if -9.99999999999999967e78 < F < 3.6e21

   1. Initial program 98.8%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 51.3

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

   5. Applied rewrites 51.3%

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

   if 3.6e21 < F

   1. Initial program 59.2%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.4%

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.112:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right), \left(x \cdot \left(B \cdot B\right)\right) \cdot 0.3333333333333333 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.112)
   (/
    (fma
     (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0))))
     (fma 0.16666666666666666 (* F (* B B)) F)
     (- (* (* x (* B B)) 0.3333333333333333) x))
    B)
   (- (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.112) {
		tmp = fma(sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), fma(0.16666666666666666, (F * (B * B)), F), (((x * (B * B)) * 0.3333333333333333) - x)) / B;
	} else {
		tmp = -(x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.112)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), fma(0.16666666666666666, Float64(F * Float64(B * B)), F), Float64(Float64(Float64(x * Float64(B * B)) * 0.3333333333333333) - x)) / B);
	else
		tmp = Float64(-Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.112], N[(N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] + N[(N[(N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 0.112000000000000002

   1. Initial program 79.9%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   5. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right), \left(x \cdot \left(B \cdot B\right)\right) \cdot 0.3333333333333333 - x\right)}{B}} \]

   if 0.112000000000000002 < B

   1. Initial program 79.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 57.4%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.112:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right), \left(x \cdot \left(B \cdot B\right)\right) \cdot 0.3333333333333333 - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 140000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5)
   (/ (- -1.0 x) B)
   (if (<= F 140000.0)
     (/ (fma F (sqrt (/ 1.0 (fma 2.0 x 2.0))) (- x)) B)
     (/ (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) (- 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 140000.0) {
		tmp = fma(F, sqrt((1.0 / fma(2.0, x, 2.0))), -x) / B;
	} else {
		tmp = fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), (1.0 - x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 140000.0)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, 2.0))), Float64(-x)) / B);
	else
		tmp = Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), Float64(1.0 - x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 140000.0], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 41.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.5 < F < 1.4e5

   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 50.4%

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

   if 1.4e5 < F

   1. Initial program 61.4%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 27.8

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

   5. Applied rewrites 27.8%

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

   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 43.6%

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

Alternative 15: 50.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 140000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42)
   (/ (- -1.0 x) B)
   (if (<= F 140000.0)
     (/ (fma F (sqrt 0.5) (- x)) B)
     (/ (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) (- 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 140000.0) {
		tmp = fma(F, sqrt(0.5), -x) / B;
	} else {
		tmp = fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), (1.0 - x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 140000.0)
		tmp = Float64(fma(F, sqrt(0.5), Float64(-x)) / B);
	else
		tmp = Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), Float64(1.0 - x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 140000.0], N[(N[(F * N[Sqrt[0.5], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 41.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.4199999999999999 < F < 1.4e5

   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.4%

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

   if 1.4e5 < F

   1. Initial program 61.4%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 27.8

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

   5. Applied rewrites 27.8%

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

   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 43.6%

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

Alternative 16: 50.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 140000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42)
   (/ (- -1.0 x) B)
   (if (<= F 140000.0) (/ (fma F (sqrt 0.5) (- x)) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 140000.0) {
		tmp = fma(F, sqrt(0.5), -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 140000.0)
		tmp = Float64(fma(F, sqrt(0.5), Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 140000.0], N[(N[(F * N[Sqrt[0.5], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 41.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.4199999999999999 < F < 1.4e5

   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.4%

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

   if 1.4e5 < F

   1. Initial program 61.4%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 43.5%

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

Alternative 17: 43.5% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.00105:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.65e-64)
   (/ (- -1.0 x) B)
   (if (<= F 0.00105)
     (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.65e-64) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.00105) {
		tmp = (x * fma(0.3333333333333333, (B * B), -1.0)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.65e-64)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.00105)
		tmp = Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.65e-64], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.00105], N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6500000000000001e-64

   1. Initial program 68.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 39.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.6500000000000001e-64 < F < 0.00104999999999999994

   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 x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 32.9%

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

   if 0.00104999999999999994 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 43.1%

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

Alternative 18: 43.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.48 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.48e-77)
   (/ (- -1.0 x) B)
   (if (<= F 1.9e-16) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.48e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-16) {
		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 <= (-1.48d-77)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.9d-16) 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 <= -1.48e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-16) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.48e-77:
		tmp = (-1.0 - x) / B
	elif F <= 1.9e-16:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.48e-77)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.9e-16)
		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 <= -1.48e-77)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.9e-16)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.48e-77], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-16], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.48000000000000002e-77

   1. Initial program 69.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 38.5%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.48000000000000002e-77 < F < 1.90000000000000006e-16

   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 32.8%

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

   if 1.90000000000000006e-16 < F

   1. Initial program 62.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 28.1

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

   5. Applied rewrites 28.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 43.2%

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

Alternative 19: 36.6% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.48 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.48e-77) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.48e-77) {
		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 <= (-1.48d-77)) 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 <= -1.48e-77) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.48e-77:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.48e-77)
		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 <= -1.48e-77)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.48e-77], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.48000000000000002e-77

   1. Initial program 69.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 38.5%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.48000000000000002e-77 < F

   1. Initial program 83.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 26.7%

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

Alternative 20: 29.4% 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 79.8%

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

3. Taylor expanded in 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. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 39.2

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

5. Applied rewrites 39.2%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 25.5%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Alternative 21: 10.6% 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 79.8%

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

3. Taylor expanded in 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. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 39.2

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

5. Applied rewrites 39.2%

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

6. Taylor expanded in F around -inf

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

8. Applied rewrites 23.3%

   \[\leadsto \frac{-1 - x}{B} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 7.4%

    \[\leadsto \frac{-1}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(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))))))