VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.7%

Time: 17.1s

Alternatives: 18

Speedup: 2.3×

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+115}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+115)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3e+77)
       (- (/ F (* (sin B) (sqrt (fma F F (fma 2.0 x 2.0))))) t_0)
       (fma (/ (/ 1.0 (sin B)) F) F (/ x (- 0.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+115) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3e+77) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, fma(2.0, x, 2.0))))) - t_0;
	} else {
		tmp = fma(((1.0 / sin(B)) / F), F, (x / (0.0 - tan(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+115)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3e+77)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, fma(2.0, x, 2.0))))) - t_0);
	else
		tmp = fma(Float64(Float64(1.0 / sin(B)) / F), F, Float64(x / Float64(0.0 - tan(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+115], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3e+77], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e115

   1. Initial program 38.6%

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

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

   5. Simplified 99.7%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. unsub-neg N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      8. +-lft-identity N/A

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

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

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

      10. tan-lowering-tan.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if -4.0000000000000001e115 < F < 2.9999999999999998e77

   1. Initial program 95.5%

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

      1. +-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.6%

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

   if 2.9999999999999998e77 < F

   1. Initial program 45.2%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sin B \cdot F}}, F, 0 - \frac{x}{\tan B}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1} \cdot F}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\frac{1}{F}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B} \cdot F}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{F \cdot \sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{F \cdot \sin B} \cdot F}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F} \cdot F}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      13. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{1}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      17. sin-lowering-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+115}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 2.0)))) t_0)
       (fma (/ (/ 1.0 (sin B)) F) F (/ x (- 0.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0;
	} else {
		tmp = fma(((1.0 / sin(B)) / F), F, (x / (0.0 - tan(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, 2.0)))) - t_0);
	else
		tmp = fma(Float64(Float64(1.0 / sin(B)) / F), F, Float64(x / Float64(0.0 - tan(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], 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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 51.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 98.4

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

   5. Simplified 98.4%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. unsub-neg N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      8. +-lft-identity N/A

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

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

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

      10. tan-lowering-tan.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   if -1.44999999999999996 < F < 1.3999999999999999

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0

      \[\leadsto \frac{F}{\color{blue}{\sin B \cdot \sqrt{2 + 2 \cdot x}}} - \frac{x}{\tan B} \]
   8. 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 99.5

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

   9. Simplified 99.5%

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

   if 1.3999999999999999 < F

   1. Initial program 57.9%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 98.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 98.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sin B \cdot F}}, F, 0 - \frac{x}{\tan B}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1} \cdot F}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\frac{1}{F}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B} \cdot F}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{F \cdot \sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{F \cdot \sin B} \cdot F}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F} \cdot F}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      13. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{1}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      17. sin-lowering-sin.f64 98.8

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

   9. Applied egg-rr 98.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1400:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 60000:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1400.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 60000.0)
       (- (/ F (* B (sqrt (fma F F 2.0)))) t_0)
       (fma (/ (/ 1.0 (sin B)) F) F (/ x (- 0.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1400.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 60000.0) {
		tmp = (F / (B * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = fma(((1.0 / sin(B)) / F), F, (x / (0.0 - tan(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1400.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 60000.0)
		tmp = Float64(Float64(F / Float64(B * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = fma(Float64(Float64(1.0 / sin(B)) / F), F, Float64(x / Float64(0.0 - tan(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1400.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 60000.0], N[(N[(F / N[(B * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1400

   1. Initial program 50.5%

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

   3. Taylor expanded in 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 99.0

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

   5. Simplified 99.0%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. unsub-neg N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      8. +-lft-identity N/A

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

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

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

      10. tan-lowering-tan.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if -1400 < F < 6e4

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 82.5%

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

   10. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. accelerator-lowering-fma.f64 82.5

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

   12. Simplified 82.5%

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

   if 6e4 < F

   1. Initial program 56.7%

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

      1. +-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. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sin B \cdot F}}, F, 0 - \frac{x}{\tan B}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1} \cdot F}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\frac{1}{F}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B} \cdot F}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{F \cdot \sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{F \cdot \sin B} \cdot F}{F}}, F, 0 - \frac{x}{\tan B}\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F}}{\sin B}} \cdot F}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{\frac{1}{F} \cdot F}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{-1}} \cdot F}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      13. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{{F}^{\left(-1 + 1\right)}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{{F}^{\color{blue}{0}}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{1}}{\sin B}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

      17. sin-lowering-sin.f64 99.7

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\sin B}}}{F}, F, 0 - \frac{x}{\tan B}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sin B}}{F}}, F, 0 - \frac{x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1400:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 60000:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{\sin B}}{F}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1520:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 62000:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1520.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 62000.0)
       (- (/ F (* B (sqrt (fma F F 2.0)))) t_0)
       (- (/ 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 <= -1520.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 62000.0) {
		tmp = (F / (B * sqrt(fma(F, F, 2.0)))) - t_0;
	} 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 <= -1520.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 62000.0)
		tmp = Float64(Float64(F / Float64(B * sqrt(fma(F, F, 2.0)))) - t_0);
	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, -1520.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 62000.0], N[(N[(F / N[(B * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1520

   1. Initial program 50.5%

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

   3. Taylor expanded in 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 99.0

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

   5. Simplified 99.0%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. unsub-neg N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      8. +-lft-identity N/A

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

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

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

      10. tan-lowering-tan.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if -1520 < F < 62000

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 82.5%

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

   10. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. accelerator-lowering-fma.f64 82.5

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

   12. Simplified 82.5%

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

   if 62000 < F

   1. Initial program 56.7%

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

      1. +-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. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{1}{F \cdot \sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      2. +-rgt-identity N/A

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

      3. +-lft-identity N/A

         \[\leadsto \frac{1}{F \cdot \sin B} \cdot F - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{1}{F \cdot \sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{F}}{\sin B}} \cdot F - \left(0 + \frac{x}{\tan B}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{F} \cdot F}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      7. inv-pow N/A

         \[\leadsto \frac{\color{blue}{{F}^{-1}} \cdot F}{\sin B} - \left(0 + \frac{x}{\tan B}\right) \]

      8. pow-plus N/A

         \[\leadsto \frac{\color{blue}{{F}^{\left(-1 + 1\right)}}}{\sin B} - \left(0 + \frac{x}{\tan B}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{{F}^{\color{blue}{0}}}{\sin B} - \left(0 + \frac{x}{\tan B}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{1}}{\sin B} - \left(0 + \frac{x}{\tan B}\right) \]

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

          \[\leadsto \color{blue}{\frac{1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      13. +-lft-identity N/A

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

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

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

      15. tan-lowering-tan.f64 99.7

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

   9. Applied egg-rr 99.7%

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

Alternative 5: 84.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1520:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+162}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1520.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7e+162)
       (- (/ F (* B (sqrt (fma F F 2.0)))) t_0)
       (fma (/ 1.0 (* F (sin B))) F (/ (- 0.0 x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1520.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7e+162) {
		tmp = (F / (B * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = fma((1.0 / (F * sin(B))), F, ((0.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1520.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7e+162)
		tmp = Float64(Float64(F / Float64(B * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = fma(Float64(1.0 / Float64(F * sin(B))), F, Float64(Float64(0.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1520.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7e+162], N[(N[(F / N[(B * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1520

   1. Initial program 50.5%

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

   3. Taylor expanded in 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 99.0

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

   5. Simplified 99.0%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. unsub-neg N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \left(0 + \frac{x}{\tan B}\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \left(0 + \frac{x}{\tan B}\right) \]

      8. +-lft-identity N/A

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

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

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

      10. tan-lowering-tan.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if -1520 < F < 7.00000000000000036e162

   1. Initial program 96.7%

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

      1. +-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 79.4%

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

   10. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. accelerator-lowering-fma.f64 79.4

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

   12. Simplified 79.4%

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

   if 7.00000000000000036e162 < F

   1. Initial program 11.3%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 91.4

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]

   10. Simplified 91.4%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1520:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+162}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+22}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e-96)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 3.1e-144)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (if (<= F 1.32e+22)
       (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B)))
       (if (<= F 1.6e+173)
         (fma (/ 1.0 (* F B)) F (/ x (- 0.0 (tan B))))
         (fma (/ 1.0 (* F (sin B))) F (/ (- 0.0 x) B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-96) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 3.1e-144) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else if (F <= 1.32e+22) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B));
	} else if (F <= 1.6e+173) {
		tmp = fma((1.0 / (F * B)), F, (x / (0.0 - tan(B))));
	} else {
		tmp = fma((1.0 / (F * sin(B))), F, ((0.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-96)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 3.1e-144)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	elseif (F <= 1.32e+22)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)));
	elseif (F <= 1.6e+173)
		tmp = fma(Float64(1.0 / Float64(F * B)), F, Float64(x / Float64(0.0 - tan(B))));
	else
		tmp = fma(Float64(1.0 / Float64(F * sin(B))), F, Float64(Float64(0.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-96], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-144], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.32e+22], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e+173], N[(N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.2000000000000003e-96

   1. Initial program 57.1%

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

      1. +-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 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 62.8%

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

   10. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 74.9

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

   12. Simplified 74.9%

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

   if -5.2000000000000003e-96 < F < 3.1000000000000001e-144

   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 54.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{x \cdot 2 + \left(F \cdot F + 2\right)}} + \left(0 - x\right)}{B}} \]

   7. Applied egg-rr 54.9%

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

   if 3.1000000000000001e-144 < F < 1.32e22

   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 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 68.3

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

   5. Simplified 68.3%

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

   if 1.32e22 < F < 1.6000000000000001e173

   1. Initial program 83.6%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.4

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]
   9. Step-by-step derivation

   10. Simplified 74.9%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]

   if 1.6000000000000001e173 < F

   1. Initial program 9.0%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 93.7

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]

   10. Simplified 93.7%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+22}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+162}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e+153)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7e+162)
     (- (/ F (* B (sqrt (fma F F 2.0)))) (/ x (tan B)))
     (fma (/ 1.0 (* F (sin B))) F (/ (- 0.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e+153) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7e+162) {
		tmp = (F / (B * sqrt(fma(F, F, 2.0)))) - (x / tan(B));
	} else {
		tmp = fma((1.0 / (F * sin(B))), F, ((0.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e+153)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7e+162)
		tmp = Float64(Float64(F / Float64(B * sqrt(fma(F, F, 2.0)))) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(1.0 / Float64(F * sin(B))), F, Float64(Float64(0.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e+153], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+162], N[(N[(F / N[(B * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.20000000000000033e153

   1. Initial program 29.8%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 79.5

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

   8. Simplified 79.5%

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

   if -8.20000000000000033e153 < F < 7.00000000000000036e162

   1. Initial program 94.5%

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

      1. +-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 79.7%

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

   10. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. accelerator-lowering-fma.f64 79.7

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

   12. Simplified 79.7%

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

   if 7.00000000000000036e162 < F

   1. Initial program 11.3%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 91.4

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]

   10. Simplified 91.4%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+162}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-117}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-109)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 1e-117)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (if (<= F 7e+172)
       (fma (/ 1.0 (* F B)) F (/ x (- 0.0 (tan B))))
       (fma (/ 1.0 (* F (sin B))) F (/ (- 0.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-109) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 1e-117) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else if (F <= 7e+172) {
		tmp = fma((1.0 / (F * B)), F, (x / (0.0 - tan(B))));
	} else {
		tmp = fma((1.0 / (F * sin(B))), F, ((0.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-109)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 1e-117)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	elseif (F <= 7e+172)
		tmp = fma(Float64(1.0 / Float64(F * B)), F, Float64(x / Float64(0.0 - tan(B))));
	else
		tmp = fma(Float64(1.0 / Float64(F * sin(B))), F, Float64(Float64(0.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-109], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-117], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7e+172], N[(N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.39999999999999989e-109

   1. Initial program 57.1%

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

      1. +-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 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 62.8%

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

   10. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 74.9

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

   12. Simplified 74.9%

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

   if -1.39999999999999989e-109 < F < 1.00000000000000003e-117

   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 56.1

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{x \cdot 2 + \left(F \cdot F + 2\right)}} + \left(0 - x\right)}{B}} \]

   7. Applied egg-rr 56.2%

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

   if 1.00000000000000003e-117 < F < 6.99999999999999955e172

   1. Initial program 89.4%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 78.1

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 78.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]
   9. Step-by-step derivation

   10. Simplified 62.0%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]

   if 6.99999999999999955e172 < F

   1. Initial program 9.0%

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

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 93.7

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]

   10. Simplified 93.7%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, 0 - \color{blue}{\frac{x}{B}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-117}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot \sin B}, F, \frac{0 - x}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8e-98)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 4.9e-118)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (fma (/ 1.0 (* F B)) F (/ x (- 0.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e-98) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 4.9e-118) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = fma((1.0 / (F * B)), F, (x / (0.0 - tan(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8e-98)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 4.9e-118)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = fma(Float64(1.0 / Float64(F * B)), F, Float64(x / Float64(0.0 - tan(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8e-98], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.9e-118], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision] * F + N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8000000000000003e-98

   1. Initial program 57.1%

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

      1. +-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 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 62.8%

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

   10. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 74.9

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

   12. Simplified 74.9%

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

   if -6.8000000000000003e-98 < F < 4.8999999999999998e-118

   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 56.1

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{x \cdot 2 + \left(F \cdot F + 2\right)}} + \left(0 - x\right)}{B}} \]

   7. Applied egg-rr 56.2%

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

   if 4.8999999999999998e-118 < F

   1. Initial program 66.5%

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

      1. +-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. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 84.2

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 84.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]
   9. Step-by-step derivation

   10. Simplified 62.7%

       \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{B}}, F, 0 - \frac{x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F \cdot B}, F, \frac{x}{0 - \tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.052:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right), \left(x \cdot \left(B \cdot B\right)\right) \cdot 0.3333333333333333\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.052)
   (/
    (-
     (fma
      (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))
      (fma 0.16666666666666666 (* F (* B B)) F)
      (* (* x (* B B)) 0.3333333333333333))
     x)
    B)
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.052) {
		tmp = (fma(sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))), fma(0.16666666666666666, (F * (B * B)), F), ((x * (B * B)) * 0.3333333333333333)) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.052)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))), fma(0.16666666666666666, Float64(F * Float64(B * B)), F), Float64(Float64(x * Float64(B * B)) * 0.3333333333333333)) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.052], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] + N[(N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0519999999999999976

   1. Initial program 67.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in B around 0

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

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

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

   7. Simplified 53.0%

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

   if 0.0519999999999999976 < B

   1. Initial program 82.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, 0 - \frac{x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}\right)}{\sin B}} + \left(0 - \frac{x}{\tan B}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{F \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F}}{\sin B} + \left(0 - \frac{x}{\tan B}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} + \left(0 - \frac{x}{\tan B}\right) \]

      5. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \frac{x}{\tan B}} \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F + 0\right) - \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]

      7. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \left(0 + \frac{x}{\tan B}\right) \]

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

         \[\leadsto \color{blue}{\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \left(0 + \frac{x}{\tan B}\right)} \]

   6. Applied egg-rr 82.4%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 53.1%

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

   10. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 48.0

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

   12. Simplified 48.0%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 62000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+154)
   (/ (- -1.0 x) B)
   (if (<= F 62000.0)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 62000.0) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 62000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 62000.0], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000007e154

   1. Initial program 29.8%

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

   3. Taylor expanded in B around 0

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

      1. /-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 27.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 27.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 56.7

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

   8. Simplified 56.7%

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

   if -2.00000000000000007e154 < F < 62000

   1. Initial program 95.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 51.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{x \cdot 2 + \left(F \cdot F + 2\right)}} + \left(0 - x\right)}{B}} \]

   7. Applied egg-rr 51.8%

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

   if 62000 < F

   1. Initial program 56.7%

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

      1. +-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. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

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

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

   4. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, 0 - \frac{x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

      3. sin-lowering-sin.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{1}{F \cdot \color{blue}{\sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F \cdot \sin B}}, F, 0 - \frac{x}{\tan B}\right) \]

   8. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 59.9

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

   10. Simplified 59.9%

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

Alternative 12: 50.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{2}{F \cdot \left(F \cdot B\right)}, \frac{1 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+154)
   (/ (- -1.0 x) B)
   (if (<= F 4.5e-42)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (fma -0.5 (/ 2.0 (* F (* F B))) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-42) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = fma(-0.5, (2.0 / (F * (F * B))), ((1.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.5e-42)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = fma(-0.5, Float64(2.0 / Float64(F * Float64(F * B))), Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-42], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(-0.5 * N[(2.0 / N[(F * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000007e154

   1. Initial program 29.8%

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

   3. Taylor expanded in B around 0

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

      1. /-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 27.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 27.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 56.7

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

   8. Simplified 56.7%

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

   if -2.00000000000000007e154 < F < 4.5e-42

   1. Initial program 95.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 52.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{x \cdot 2 + \left(F \cdot F + 2\right)}} + \left(0 - x\right)}{B}} \]

   7. Applied egg-rr 52.6%

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

   if 4.5e-42 < F

   1. Initial program 60.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 28.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   6. Taylor expanded in F around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. --lowering--.f64 43.2

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

   8. Simplified 43.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 44.3%

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

Alternative 13: 43.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.22:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{2}{F \cdot \left(F \cdot B\right)}, \frac{1 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.22)
   (/ (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) (- -1.0 x)) B)
   (if (<= F 4.5e-42)
     (/ (- 0.0 x) B)
     (fma -0.5 (/ 2.0 (* F (* F B))) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.22) {
		tmp = fma(0.5, (fma(2.0, x, 2.0) / (F * F)), (-1.0 - x)) / B;
	} else if (F <= 4.5e-42) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = fma(-0.5, (2.0 / (F * (F * B))), ((1.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.22)
		tmp = Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), Float64(-1.0 - x)) / B);
	elseif (F <= 4.5e-42)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = fma(-0.5, Float64(2.0 / Float64(F * Float64(F * B))), Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.22], N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-42], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(-0.5 * N[(2.0 / N[(F * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.220000000000000001

   1. Initial program 51.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 37.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   6. Taylor expanded in F around -inf

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

      1. sub-neg N/A

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

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

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

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

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 55.1

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

   8. Simplified 55.1%

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

   if -0.220000000000000001 < F < 4.5e-42

   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. 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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 52.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

      3. --lowering--.f64 33.2

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   8. Simplified 33.2%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 4.5e-42 < F

   1. Initial program 60.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 28.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   6. Taylor expanded in F around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. --lowering--.f64 43.2

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

   8. Simplified 43.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 44.3%

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

Alternative 14: 43.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{2}{F \cdot \left(F \cdot B\right)}, \frac{1 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.1e-36)
   (/ (- -1.0 x) B)
   (if (<= F 4.5e-42)
     (/ (- 0.0 x) B)
     (fma -0.5 (/ 2.0 (* F (* F B))) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.1e-36) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-42) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = fma(-0.5, (2.0 / (F * (F * B))), ((1.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.1e-36)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.5e-42)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = fma(-0.5, Float64(2.0 / Float64(F * Float64(F * B))), Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.1e-36], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-42], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(-0.5 * N[(2.0 / N[(F * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.09999999999999973e-36

   1. Initial program 53.8%

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

   3. Taylor expanded in B around 0

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

      1. /-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 39.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 53.6

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

   8. Simplified 53.6%

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

   if -5.09999999999999973e-36 < F < 4.5e-42

   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. 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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 51.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

      3. --lowering--.f64 33.5

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   8. Simplified 33.5%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 4.5e-42 < F

   1. Initial program 60.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 28.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   6. Taylor expanded in F around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. --lowering--.f64 43.2

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

   8. Simplified 43.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 44.3%

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

Alternative 15: 43.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.4e-36)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e-115) (/ (- 0.0 x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e-36) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-115) {
		tmp = (0.0 - 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 <= (-5.4d-36)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.8d-115) then
        tmp = (0.0d0 - 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 <= -5.4e-36) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-115) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.4e-36:
		tmp = (-1.0 - x) / B
	elif F <= 2.8e-115:
		tmp = (0.0 - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.4e-36)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e-115)
		tmp = Float64(Float64(0.0 - 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 <= -5.4e-36)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.8e-115)
		tmp = (0.0 - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.4e-36], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-115], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.40000000000000015e-36

   1. Initial program 53.8%

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

   3. Taylor expanded in B around 0

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

      1. /-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 39.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 53.6

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

   8. Simplified 53.6%

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

   if -5.40000000000000015e-36 < F < 2.79999999999999987e-115

   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 54.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

      3. --lowering--.f64 37.5

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   8. Simplified 37.5%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 2.79999999999999987e-115 < F

   1. Initial program 66.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 29.0

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 38.9

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

   8. Simplified 38.9%

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

Alternative 16: 36.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.5e-36) (/ (- -1.0 x) B) (/ (- 0.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-36) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (0.0 - 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 <= (-7.5d-36)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = (0.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-36) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (0.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.5e-36:
		tmp = (-1.0 - x) / B
	else:
		tmp = (0.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.5e-36)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(0.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.5e-36)
		tmp = (-1.0 - x) / B;
	else
		tmp = (0.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.5e-36], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -7.49999999999999972e-36

   1. Initial program 53.8%

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

   3. Taylor expanded in B around 0

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

      1. /-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 39.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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 53.6

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

   8. Simplified 53.6%

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

   if -7.49999999999999972e-36 < F

   1. Initial program 80.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

      15. --lowering--.f64 40.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   5. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

   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-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

      3. --lowering--.f64 25.2

         \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   8. Simplified 25.2%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.9% accurate, 24.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1 - x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- -1.0 x) B))

C

double code(double F, double B, double x) {
	return (-1.0 - x) / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((-1.0d0) - x) / b
end function

Java

public static double code(double F, double B, double x) {
	return (-1.0 - x) / B;
}

Python

def code(F, B, x):
	return (-1.0 - x) / B

Julia

function code(F, B, x)
	return Float64(Float64(-1.0 - x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = (-1.0 - x) / B;
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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-sub0 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   15. --lowering--.f64 39.9

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

5. Simplified 39.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

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 28.6

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

8. Simplified 28.6%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]
9. Add Preprocessing

Alternative 18: 10.5% 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 71.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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-sub0 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

   15. --lowering--.f64 39.9

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{0 - x}\right)}{B} \]

5. Simplified 39.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0 - x\right)}{B}} \]

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 28.6

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

8. Simplified 28.6%

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

9. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 12.3

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

11. Simplified 12.3%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))))