VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.6%

Time: 16.5s

Alternatives: 20

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: 76.7% 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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+156)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 1000000.0)
     (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+156) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 1000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+156)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 1000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+156], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1000000.0], 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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.9999999999999998e155

   1. Initial program 24.3%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. sub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. sin-lowering-sin.f64 99.8

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

   5. Simplified 99.8%

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

   if -9.9999999999999998e155 < F < 1e6

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.7%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 99.6

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

   6. Applied egg-rr 99.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 99.7%

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

   if 1e6 < F

   1. Initial program 55.0%

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

   3. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 500000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+151)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 500000000.0)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+151) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 500000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+151)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 500000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + 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, -2e+151], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 500000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000003e151

   1. Initial program 24.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 38.8%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 38.8

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

   6. Applied egg-rr 38.8%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 38.8%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 99.7

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

   11. Simplified 99.7%

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

   if -2.00000000000000003e151 < F < 5e8

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.7%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 99.6

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

   6. Applied egg-rr 99.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 99.7%

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

   if 5e8 < F

   1. Initial program 55.0%

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

   3. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 500000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -190:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -190.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.35)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 2.0)))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -190.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.35) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -190.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.35)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + 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, -190.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.35], 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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -190

   1. Initial program 55.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 65.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 65.6

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

   6. Applied egg-rr 65.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 65.7%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 99.8

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

   11. Simplified 99.8%

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

   if -190 < F < 1.3500000000000001

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 99.6

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

   6. Applied egg-rr 99.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in F around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sin-lowering-sin.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 99.7

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

   11. Simplified 99.7%

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

   if 1.3500000000000001 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -190:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;t\_0 + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.45e-12)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 4.6e-57)
       (+ t_0 (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.45e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.6e-57) {
		tmp = t_0 + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.45e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.6e-57)
		tmp = Float64(t_0 + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(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]}, If[LessEqual[F, -1.45e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-57], N[(t$95$0 + 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 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4500000000000001e-12

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 66.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 66.6

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

   6. Applied egg-rr 66.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 66.7%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 99.8

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

   11. Simplified 99.8%

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

   if -1.4500000000000001e-12 < F < 4.6e-57

   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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. /-lowering-/.f64 85.9

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

   5. Simplified 85.9%

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

   if 4.6e-57 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 94.7

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

   5. Simplified 94.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45e-12)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.6e-57)
       (+
        (* x (/ -1.0 (tan B)))
        (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.6e-57) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.6e-57)
		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 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.6e-57], 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[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4500000000000001e-12

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 66.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 66.6

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

   6. Applied egg-rr 66.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 66.7%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 99.8

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

   11. Simplified 99.8%

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

   if -1.4500000000000001e-12 < F < 4.6e-57

   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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. /-lowering-/.f64 85.9

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

   5. Simplified 85.9%

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

   if 4.6e-57 < F

   1. Initial program 59.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 78.8%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 78.8

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

   6. Applied egg-rr 78.8%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 78.9%

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

   9. Taylor expanded in F around inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 94.7

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

   11. Simplified 94.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{+60}:\\ \;\;\;\;t\_0 + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.45e-12)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 3.05e+60)
       (+ t_0 (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
       (if (<= F 3.5e+123)
         (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.45e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3.05e+60) {
		tmp = t_0 + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (F <= 3.5e+123) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.45e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3.05e+60)
		tmp = Float64(t_0 + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (F <= 3.5e+123)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.05e+60], N[(t$95$0 + 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], If[LessEqual[F, 3.5e+123], 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 / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.4500000000000001e-12

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 66.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 66.6

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

   6. Applied egg-rr 66.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 66.7%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 N/A

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

       2. sin-lowering-sin.f64 99.8

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

   11. Simplified 99.8%

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

   if -1.4500000000000001e-12 < F < 3.05e60

   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

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. /-lowering-/.f64 84.1

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

   5. Simplified 84.1%

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

   if 3.05e60 < F < 3.5e123

   1. Initial program 71.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 99.6

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

   6. Applied egg-rr 99.6%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in B around 0

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

   11. Simplified 92.7%

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

   if 3.5e123 < F

   1. Initial program 38.7%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 61.2

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

   5. Simplified 61.2%

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

      1. clear-num N/A

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

      2. distribute-neg-frac N/A

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

      3. metadata-eval N/A

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

      4. associate-/l/ N/A

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

      5. tan-quot N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. tan-lowering-tan.f64 61.3

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

   7. Applied egg-rr 61.3%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. tan-lowering-tan.f64 61.3

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

   9. Applied egg-rr 61.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -6.2e-93)
     t_0
     (if (<= x 1.2e-11)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -6.2e-93) {
		tmp = t_0;
	} else if (x <= 1.2e-11) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / 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 <= -6.2e-93)
		tmp = t_0;
	elseif (x <= 1.2e-11)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / 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, -6.2e-93], t$95$0, If[LessEqual[x, 1.2e-11], 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 / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999999e-93 or 1.2000000000000001e-11 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 90.8

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

   5. Simplified 90.8%

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

      1. associate-/l* N/A

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

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

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

      3. clear-num N/A

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

      4. tan-quot N/A

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

      5. div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. tan-lowering-tan.f64 90.9

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

   7. Applied egg-rr 90.9%

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

   if -6.19999999999999999e-93 < x < 1.2000000000000001e-11

   1. Initial program 70.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 72.8%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. inv-pow N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. accelerator-lowering-fma.f64 72.8

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

   6. Applied egg-rr 72.8%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

   8. Applied egg-rr 72.9%

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

   9. Taylor expanded in B around 0

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

   11. Simplified 62.1%

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

Alternative 8: 71.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.1e-107)
     t_0
     (if (<= x 1.1e-11) (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.1e-107) {
		tmp = t_0;
	} else if (x <= 1.1e-11) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / 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 <= -1.1e-107)
		tmp = t_0;
	elseif (x <= 1.1e-11)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / 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, -1.1e-107], t$95$0, If[LessEqual[x, 1.1e-11], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000006e-107 or 1.1000000000000001e-11 < x

   1. Initial program 78.0%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 89.6

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

   5. Simplified 89.6%

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

      1. associate-/l* N/A

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

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

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

      3. clear-num N/A

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

      4. tan-quot N/A

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

      5. div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. tan-lowering-tan.f64 89.8

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

   7. Applied egg-rr 89.8%

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

   if -1.10000000000000006e-107 < x < 1.1000000000000001e-11

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. sin-lowering-sin.f64 54.3

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

   5. Simplified 54.3%

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

Alternative 9: 56.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(F \cdot \left(B \cdot B\right), 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, t\_0, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0))))))
   (if (<= B 1.4)
     (/
      (fma
       (* B B)
       (fma
        t_0
        (fma (* F (* B B)) 0.019444444444444445 (* F 0.16666666666666666))
        (* x (fma 0.022222222222222223 (* B B) 0.3333333333333333)))
       (fma F t_0 (- x)))
      B)
     (/ (- x) (tan 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 (B <= 1.4) {
		tmp = fma((B * B), fma(t_0, fma((F * (B * B)), 0.019444444444444445, (F * 0.16666666666666666)), (x * fma(0.022222222222222223, (B * B), 0.3333333333333333))), fma(F, t_0, -x)) / B;
	} else {
		tmp = -x / tan(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 (B <= 1.4)
		tmp = Float64(fma(Float64(B * B), fma(t_0, fma(Float64(F * Float64(B * B)), 0.019444444444444445, Float64(F * 0.16666666666666666)), Float64(x * fma(0.022222222222222223, Float64(B * B), 0.3333333333333333))), fma(F, t_0, Float64(-x))) / B);
	else
		tmp = Float64(Float64(-x) / tan(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[B, 1.4], N[(N[(N[(B * B), $MachinePrecision] * N[(t$95$0 * N[(N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] * 0.019444444444444445 + N[(F * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.022222222222222223 * N[(B * B), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * t$95$0 + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.3999999999999999

   1. Initial program 72.0%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}} \]

   if 1.3999999999999999 < B

   1. Initial program 81.4%

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

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/l* N/A

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

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

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

      3. clear-num N/A

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

      4. tan-quot N/A

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

      5. div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. tan-lowering-tan.f64 58.6

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

   7. Applied egg-rr 58.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(F \cdot \left(B \cdot B\right), 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, x \cdot \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.4)
   (/
    (fma
     (* B B)
     (* x (fma (* B B) 0.022222222222222223 0.3333333333333333))
     (fma F (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (- x)))
    B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.4) {
		tmp = fma((B * B), (x * fma((B * B), 0.022222222222222223, 0.3333333333333333)), fma(F, sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), -x)) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.4)
		tmp = Float64(fma(Float64(B * B), Float64(x * fma(Float64(B * B), 0.022222222222222223, 0.3333333333333333)), fma(F, sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), Float64(-x))) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.4], N[(N[(N[(B * B), $MachinePrecision] * N[(x * N[(N[(B * B), $MachinePrecision] * 0.022222222222222223 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(F * N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.3999999999999999

   1. Initial program 72.0%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)}, \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)}, \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, x \cdot \color{blue}{\left(\frac{1}{45} \cdot {B}^{2} + \frac{1}{3}\right)}, \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, x \cdot \left(\color{blue}{{B}^{2} \cdot \frac{1}{45}} + \frac{1}{3}\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, x \cdot \color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{45}, \frac{1}{3}\right)}, \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, x \cdot \mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{1}{45}, \frac{1}{3}\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{neg}\left(x\right)\right)\right)}{B} \]

      6. *-lowering-*.f64 53.5

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, x \cdot \mathsf{fma}\left(\color{blue}{B \cdot B}, 0.022222222222222223, 0.3333333333333333\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B} \]

   8. Simplified 53.5%

      \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \color{blue}{x \cdot \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right)}, \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B} \]

   if 1.3999999999999999 < B

   1. Initial program 81.4%

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

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/l* N/A

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

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

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

      3. clear-num N/A

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

      4. tan-quot N/A

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

      5. div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. tan-lowering-tan.f64 58.6

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

   7. Applied egg-rr 58.6%

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

Alternative 11: 51.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.019444444444444445, \mathsf{fma}\left(x, \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right), -0.16666666666666666\right)\right), -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (/
    (-
     (fma
      (* B B)
      (fma
       (* B B)
       -0.019444444444444445
       (fma
        x
        (fma (* B B) 0.022222222222222223 0.3333333333333333)
        -0.16666666666666666))
      -1.0)
     x)
    B)
   (if (<= F 1000000.0)
     (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) B) (/ x B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = (fma((B * B), fma((B * B), -0.019444444444444445, fma(x, fma((B * B), 0.022222222222222223, 0.3333333333333333), -0.16666666666666666)), -1.0) - x) / B;
	} else if (F <= 1000000.0) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(fma(Float64(B * B), fma(Float64(B * B), -0.019444444444444445, fma(x, fma(Float64(B * B), 0.022222222222222223, 0.3333333333333333), -0.16666666666666666)), -1.0) - x) / B);
	elseif (F <= 1000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * -0.019444444444444445 + N[(x * N[(N[(B * B), $MachinePrecision] * 0.022222222222222223 + 0.3333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1000000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 24.3%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{{B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - 1\right) - x}}{B} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - 1\right) - x}}{B} \]

   8. Simplified 59.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.019444444444444445, \mathsf{fma}\left(x, \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right), -0.16666666666666666\right)\right), -1\right) - x}}{B} \]

   if -1.35000000000000003e154 < F < 1e6

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 40.4

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

   5. Simplified 40.4%

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

      1. unsub-neg N/A

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

      2. div-sub N/A

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

      3. --lowering--.f64 N/A

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

   7. Applied egg-rr 40.4%

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

   if 1e6 < F

   1. Initial program 55.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 47.6

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

   5. Simplified 47.6%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 57.6

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

   8. Simplified 57.6%

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

Alternative 12: 50.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.019444444444444445, \mathsf{fma}\left(x, \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right), -0.16666666666666666\right)\right), -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.1e+153)
   (/
    (-
     (fma
      (* B B)
      (fma
       (* B B)
       -0.019444444444444445
       (fma
        x
        (fma (* B B) 0.022222222222222223 0.3333333333333333)
        -0.16666666666666666))
      -1.0)
     x)
    B)
   (if (<= F 1e+29)
     (/ (fma F (sqrt (/ 1.0 (fma x 2.0 (fma F F 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 <= -6.1e+153) {
		tmp = (fma((B * B), fma((B * B), -0.019444444444444445, fma(x, fma((B * B), 0.022222222222222223, 0.3333333333333333), -0.16666666666666666)), -1.0) - x) / B;
	} else if (F <= 1e+29) {
		tmp = fma(F, sqrt((1.0 / fma(x, 2.0, fma(F, F, 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 <= -6.1e+153)
		tmp = Float64(Float64(fma(Float64(B * B), fma(Float64(B * B), -0.019444444444444445, fma(x, fma(Float64(B * B), 0.022222222222222223, 0.3333333333333333), -0.16666666666666666)), -1.0) - x) / B);
	elseif (F <= 1e+29)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.1e+153], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * -0.019444444444444445 + N[(x * N[(N[(B * B), $MachinePrecision] * 0.022222222222222223 + 0.3333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e+29], 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[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.0999999999999998e153

   1. Initial program 24.3%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{{B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - 1\right) - x}}{B} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\left(\frac{-7}{360} \cdot {B}^{2} + x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {B}^{2}\right)\right) - \frac{1}{6}\right) - 1\right) - x}}{B} \]

   8. Simplified 59.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.019444444444444445, \mathsf{fma}\left(x, \mathsf{fma}\left(B \cdot B, 0.022222222222222223, 0.3333333333333333\right), -0.16666666666666666\right)\right), -1\right) - x}}{B} \]

   if -6.0999999999999998e153 < F < 9.99999999999999914e28

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 40.7

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

   5. Simplified 40.7%

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

   if 9.99999999999999914e28 < F

   1. Initial program 52.6%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 47.4

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

   5. Simplified 47.4%

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

   6. Taylor expanded in F around inf

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 57.9

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

   8. Simplified 57.9%

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

Alternative 13: 43.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 310000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e-68)
   (/ (- -1.0 x) B)
   (if (<= F -5.5e-127)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 310000.0)
       (/ (- 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 <= -1.35e-68) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -5.5e-127) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 310000.0) {
		tmp = -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 <= -1.35e-68)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -5.5e-127)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 310000.0)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e-68], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -5.5e-127], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 310000.0], N[((-x) / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3500000000000001e-68

   1. Initial program 63.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 34.3

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

   5. Simplified 34.3%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 47.6

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

   8. Simplified 47.6%

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

   if -1.3500000000000001e-68 < F < -5.50000000000000036e-127

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 68.1

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

   8. Simplified 68.1%

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

   9. Taylor expanded in F around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 68.1

          \[\leadsto \frac{F \cdot \color{blue}{\sqrt{0.5}}}{B} \]

   11. Simplified 68.1%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} \]

   if -5.50000000000000036e-127 < F < 3.1e5

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 36.0

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

   5. Simplified 36.0%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 29.2

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

   8. Simplified 29.2%

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

   if 3.1e5 < F

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 48.3

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

   5. Simplified 48.3%

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

   6. Taylor expanded in F around inf

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 58.1

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

   8. Simplified 58.1%

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

Alternative 14: 50.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -190:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 310000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -190.0)
   (/ (- -1.0 x) B)
   (if (<= F 310000.0)
     (/ (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 <= -190.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 310000.0) {
		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 <= -190.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 310000.0)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, 2.0))), Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -190.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 310000.0], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -190

   1. Initial program 55.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 28.5

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

   5. Simplified 28.5%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 47.2

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

   8. Simplified 47.2%

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

   if -190 < F < 3.1e5

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in F around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*l/ N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. neg-lowering-neg.f64 41.5

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

   8. Simplified 41.5%

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

   if 3.1e5 < F

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 48.3

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

   5. Simplified 48.3%

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

   6. Taylor expanded in F around inf

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 58.1

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

   8. Simplified 58.1%

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

Alternative 15: 43.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-68)
   (/ (- -1.0 x) B)
   (if (<= F -2.9e-126)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 2.1e-121) (/ (- x) B) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-68) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.9e-126) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 2.1e-121) {
		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 <= (-4d-68)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-2.9d-126)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 2.1d-121) 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 <= -4e-68) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.9e-126) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 2.1e-121) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e-68:
		tmp = (-1.0 - x) / B
	elif F <= -2.9e-126:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 2.1e-121:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-68)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -2.9e-126)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 2.1e-121)
		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 <= -4e-68)
		tmp = (-1.0 - x) / B;
	elseif (F <= -2.9e-126)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 2.1e-121)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-68], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -2.9e-126], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.1e-121], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.00000000000000027e-68

   1. Initial program 63.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 34.3

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

   5. Simplified 34.3%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 47.6

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

   8. Simplified 47.6%

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

   if -4.00000000000000027e-68 < F < -2.89999999999999988e-126

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 68.1

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

   8. Simplified 68.1%

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

   9. Taylor expanded in F around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 68.1

          \[\leadsto \frac{F \cdot \color{blue}{\sqrt{0.5}}}{B} \]

   11. Simplified 68.1%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} \]

   if -2.89999999999999988e-126 < F < 2.0999999999999999e-121

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 40.3

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

   5. Simplified 40.3%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 33.1

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

   8. Simplified 33.1%

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

   if 2.0999999999999999e-121 < F

   1. Initial program 63.3%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 49.4

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

   8. Simplified 49.4%

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

Alternative 16: 36.1% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 360000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= F -9e-183)
     (/ (- -1.0 x) B)
     (if (<= F 360000.0) t_0 (if (<= F 2.5e+199) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -9e-183) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 360000.0) {
		tmp = t_0;
	} else if (F <= 2.5e+199) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (f <= (-9d-183)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 360000.0d0) then
        tmp = t_0
    else if (f <= 2.5d+199) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -9e-183) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 360000.0) {
		tmp = t_0;
	} else if (F <= 2.5e+199) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if F <= -9e-183:
		tmp = (-1.0 - x) / B
	elif F <= 360000.0:
		tmp = t_0
	elif F <= 2.5e+199:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (F <= -9e-183)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 360000.0)
		tmp = t_0;
	elseif (F <= 2.5e+199)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (F <= -9e-183)
		tmp = (-1.0 - x) / B;
	elseif (F <= 360000.0)
		tmp = t_0;
	elseif (F <= 2.5e+199)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[F, -9e-183], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 360000.0], t$95$0, If[LessEqual[F, 2.5e+199], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.99999999999999942e-183

   1. Initial program 70.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 37.2

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

   5. Simplified 37.2%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 43.3

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

   8. Simplified 43.3%

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

   if -8.99999999999999942e-183 < F < 3.6e5 or 2.4999999999999999e199 < F

   1. Initial program 81.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 38.8

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

   5. Simplified 38.8%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 33.3

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

   8. Simplified 33.3%

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

   if 3.6e5 < F < 2.4999999999999999e199

   1. Initial program 66.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 31.2

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

   8. Simplified 31.2%

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

   9. Taylor expanded in F around inf

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

       1. /-lowering-/.f64 38.1

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

   11. Simplified 38.1%

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

Alternative 17: 42.1% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-183)
   (/ (- -1.0 x) B)
   (if (<= F 2.1e-121) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-183) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.1e-121) {
		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 <= (-9d-183)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.1d-121) 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 <= -9e-183) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.1e-121) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-183:
		tmp = (-1.0 - x) / B
	elif F <= 2.1e-121:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-183)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.1e-121)
		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 <= -9e-183)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.1e-121)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-183], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.1e-121], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.99999999999999942e-183

   1. Initial program 70.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 37.2

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

   5. Simplified 37.2%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 43.3

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

   8. Simplified 43.3%

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

   if -8.99999999999999942e-183 < F < 2.0999999999999999e-121

   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 41.1

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

   5. Simplified 41.1%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 32.6

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

   8. Simplified 32.6%

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

   if 2.0999999999999999e-121 < F

   1. Initial program 63.3%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 49.4

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

   8. Simplified 49.4%

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

Alternative 18: 30.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -4e-217) t_0 (if (<= x 5.6e-27) (/ 1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -4e-217) {
		tmp = t_0;
	} else if (x <= 5.6e-27) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (x <= (-4d-217)) then
        tmp = t_0
    else if (x <= 5.6d-27) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -4e-217) {
		tmp = t_0;
	} else if (x <= 5.6e-27) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -4e-217:
		tmp = t_0
	elif x <= 5.6e-27:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -4e-217)
		tmp = t_0;
	elseif (x <= 5.6e-27)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -4e-217)
		tmp = t_0;
	elseif (x <= 5.6e-27)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -4e-217], t$95$0, If[LessEqual[x, 5.6e-27], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000033e-217 or 5.5999999999999999e-27 < x

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 43.3

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

   5. Simplified 43.3%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 36.6

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

   8. Simplified 36.6%

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

   if -4.00000000000000033e-217 < x < 5.5999999999999999e-27

   1. Initial program 67.3%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 33.2

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

   5. Simplified 33.2%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 27.7

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

   8. Simplified 27.7%

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

   9. Taylor expanded in F around inf

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

       1. /-lowering-/.f64 21.8

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

   11. Simplified 21.8%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 17.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{-282}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e-282) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e-282) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / 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 <= (-3.05d-282)) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e-282) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.05e-282:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.05e-282)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.05e-282)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.05e-282], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.0499999999999999e-282

   1. Initial program 75.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 40.9

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

   5. Simplified 40.9%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 12.9

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

   8. Simplified 12.9%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 17.4

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

   11. Simplified 17.4%

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

   if -3.0499999999999999e-282 < F

   1. Initial program 74.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 39.6

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

   5. Simplified 39.6%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 16.0

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

   8. Simplified 16.0%

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

   9. Taylor expanded in F around inf

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

       1. /-lowering-/.f64 18.0

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

   11. Simplified 18.0%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 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 74.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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. /-lowering-/.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. accelerator-lowering-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. sqrt-lowering-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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. neg-lowering-neg.f64 40.2

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

5. Simplified 40.2%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. accelerator-lowering-fma.f64 14.6

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

8. Simplified 14.6%

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

9. Taylor expanded in F around -inf

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

    1. /-lowering-/.f64 9.5

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

11. Simplified 9.5%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))))