Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 97.6%

Time: 11.0s

Alternatives: 10

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - z\_m \cdot t\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;t\_1 \leq 10^{+274}:\\ \;\;\;\;\frac{x \cdot 2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z\_m}}{y - t}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* y z_m) (* z_m t))))
   (*
    z_s
    (if (<= t_1 (- INFINITY))
      (* (/ x z_m) (/ 2.0 (- y t)))
      (if (<= t_1 1e+274) (/ (* x 2.0) t_1) (/ (* x (/ 2.0 z_m)) (- y t)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else if (t_1 <= 1e+274) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = (x * (2.0 / z_m)) / (y - t);
	}
	return z_s * tmp;
}

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else if (t_1 <= 1e+274) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = (x * (2.0 / z_m)) / (y - t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (y * z_m) - (z_m * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / z_m) * (2.0 / (y - t))
	elif t_1 <= 1e+274:
		tmp = (x * 2.0) / t_1
	else:
		tmp = (x * (2.0 / z_m)) / (y - t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(z_m * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	elseif (t_1 <= 1e+274)
		tmp = Float64(Float64(x * 2.0) / t_1);
	else
		tmp = Float64(Float64(x * Float64(2.0 / z_m)) / Float64(y - t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (y * z_m) - (z_m * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / z_m) * (2.0 / (y - t));
	elseif (t_1 <= 1e+274)
		tmp = (x * 2.0) / t_1;
	else
		tmp = (x * (2.0 / z_m)) / (y - t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+274], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(x * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 58.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 58.4%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 9.99999999999999921e273

   1. Initial program 98.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Add Preprocessing

   if 9.99999999999999921e273 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 64.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 79.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 64.4%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 64.4%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 64.4%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 79.2%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]
   7. Step-by-step derivation

      1. associate-*l/ 79.2%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{z \cdot \left(y - t\right)}} \]

      2. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]

      3. frac-2neg 99.8%

         \[\leadsto \frac{2}{z} \cdot \color{blue}{\frac{-x}{-\left(y - t\right)}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{z} \cdot \left(-x\right)}{-\left(y - t\right)}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{-\color{blue}{\left(y + \left(-t\right)\right)}} \]

      6. distribute-neg-in 99.9%

         \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\color{blue}{\left(-y\right) + \left(-\left(-t\right)\right)}} \]

      7. add-sqr-sqrt 40.7%

         \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \left(-\color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right)} \]

      8. sqrt-unprod 79.1%

         \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \left(-\color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}\right)} \]

      9. sqr-neg 79.1%

         \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \left(-\sqrt{\color{blue}{t \cdot t}}\right)} \]

      10. sqrt-unprod 45.2%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \left(-\color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)} \]

      11. add-sqr-sqrt 75.2%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \left(-\color{blue}{t}\right)} \]

      12. add-sqr-sqrt 30.1%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}} \]

      13. sqrt-unprod 82.4%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}} \]

      14. sqr-neg 82.4%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \sqrt{\color{blue}{t \cdot t}}} \]

      15. sqrt-unprod 59.2%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \color{blue}{\sqrt{t} \cdot \sqrt{t}}} \]

      16. add-sqr-sqrt 99.9%

          \[\leadsto \frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + \color{blue}{t}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{z} \cdot \left(-x\right)}{\left(-y\right) + t}} \]
   9. Step-by-step derivation

      1. frac-2neg 99.9%

         \[\leadsto \color{blue}{\frac{-\frac{2}{z} \cdot \left(-x\right)}{-\left(\left(-y\right) + t\right)}} \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\left(-\frac{2}{z} \cdot \left(-x\right)\right) \cdot \frac{1}{-\left(\left(-y\right) + t\right)}} \]

      3. distribute-rgt-neg-out 99.8%

         \[\leadsto \left(-\color{blue}{\left(-\frac{2}{z} \cdot x\right)}\right) \cdot \frac{1}{-\left(\left(-y\right) + t\right)} \]

      4. remove-double-neg 99.8%

         \[\leadsto \color{blue}{\left(\frac{2}{z} \cdot x\right)} \cdot \frac{1}{-\left(\left(-y\right) + t\right)} \]

      5. distribute-neg-in 99.8%

         \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{\left(-\left(-y\right)\right) + \left(-t\right)}} \]

      6. add-sqr-sqrt 44.4%

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

      7. sqrt-unprod 92.8%

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

      8. sqr-neg 92.8%

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

      9. sqrt-unprod 51.8%

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

      10. add-sqr-sqrt 92.7%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\left(-\color{blue}{y}\right) + \left(-t\right)} \]

      11. sub-neg 92.7%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{\left(-y\right) - t}} \]

      12. add-sqr-sqrt 40.9%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}} - t} \]

      13. sqrt-unprod 96.3%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}} - t} \]

      14. sqr-neg 96.3%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\sqrt{\color{blue}{y \cdot y}} - t} \]

      15. sqrt-unprod 55.4%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{\sqrt{y} \cdot \sqrt{y}} - t} \]

      16. add-sqr-sqrt 99.8%

          \[\leadsto \left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{\color{blue}{y} - t} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\left(\frac{2}{z} \cdot x\right) \cdot \frac{1}{y - t}} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \color{blue}{\frac{\left(\frac{2}{z} \cdot x\right) \cdot 1}{y - t}} \]

       2. *-rgt-identity 99.9%

          \[\leadsto \frac{\color{blue}{\frac{2}{z} \cdot x}}{y - t} \]

       3. *-commutative 99.9%

          \[\leadsto \frac{\color{blue}{x \cdot \frac{2}{z}}}{y - t} \]

   12. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{x \cdot \frac{2}{z}}{y - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{+274}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-52} \lor \neg \left(y \leq 1.15 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z\_m \cdot t\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= y -3.5e-52) (not (<= y 1.15e+15)))
    (* x (/ (/ 2.0 z_m) y))
    (/ x (* (* z_m t) -0.5)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -3.5e-52) || !(y <= 1.15e+15)) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = x / ((z_m * t) * -0.5);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-3.5d-52)) .or. (.not. (y <= 1.15d+15))) then
        tmp = x * ((2.0d0 / z_m) / y)
    else
        tmp = x / ((z_m * t) * (-0.5d0))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -3.5e-52) || !(y <= 1.15e+15)) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = x / ((z_m * t) * -0.5);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (y <= -3.5e-52) or not (y <= 1.15e+15):
		tmp = x * ((2.0 / z_m) / y)
	else:
		tmp = x / ((z_m * t) * -0.5)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((y <= -3.5e-52) || !(y <= 1.15e+15))
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / y));
	else
		tmp = Float64(x / Float64(Float64(z_m * t) * -0.5));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((y <= -3.5e-52) || ~((y <= 1.15e+15)))
		tmp = x * ((2.0 / z_m) / y);
	else
		tmp = x / ((z_m * t) * -0.5);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[y, -3.5e-52], N[Not[LessEqual[y, 1.15e+15]], $MachinePrecision]], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z$95$m * t), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e-52 or 1.15e15 < y

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 94.8%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 91.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 91.4%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 91.4%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 94.7%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

   7. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{\frac{2}{y \cdot z}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative 81.8%

         \[\leadsto \frac{2}{\color{blue}{z \cdot y}} \cdot x \]

   9. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{2}{z \cdot y}} \cdot x \]

   10. Taylor expanded in z around 0 81.8%

       \[\leadsto \color{blue}{\frac{2}{y \cdot z}} \cdot x \]
   11. Step-by-step derivation

       1. associate-/l/ 83.0%

          \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y}} \cdot x \]

   12. Simplified 83.0%

       \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y}} \cdot x \]

   if -3.5e-52 < y < 1.15e15

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 92.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.5%

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

      1. associate-*r* 72.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(-1 \cdot t\right) \cdot z}} \]

      2. neg-mul-1 72.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(-t\right)} \cdot z} \]

      3. *-commutative 72.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(-t\right)}} \]

   7. Simplified 72.5%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(-t\right)}} \]
   8. Step-by-step derivation

      1. distribute-rgt-neg-out 72.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-z \cdot t}} \]

      2. distribute-frac-neg2 72.5%

         \[\leadsto \color{blue}{-\frac{x \cdot 2}{z \cdot t}} \]

      3. distribute-frac-neg 72.5%

         \[\leadsto \color{blue}{\frac{-x \cdot 2}{z \cdot t}} \]

      4. *-commutative 72.5%

         \[\leadsto \frac{-\color{blue}{2 \cdot x}}{z \cdot t} \]

      5. distribute-lft-neg-in 72.5%

         \[\leadsto \frac{\color{blue}{\left(-2\right) \cdot x}}{z \cdot t} \]

      6. metadata-eval 72.5%

         \[\leadsto \frac{\color{blue}{-2} \cdot x}{z \cdot t} \]

      7. times-frac 70.9%

         \[\leadsto \color{blue}{\frac{-2}{z} \cdot \frac{x}{t}} \]

   9. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{-2}{z} \cdot \frac{x}{t}} \]
   10. Step-by-step derivation

       1. clear-num 70.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{-2}}} \cdot \frac{x}{t} \]

       2. frac-2neg 70.9%

          \[\leadsto \frac{1}{\frac{z}{-2}} \cdot \color{blue}{\frac{-x}{-t}} \]

       3. frac-times 72.5%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(-x\right)}{\frac{z}{-2} \cdot \left(-t\right)}} \]

       4. *-un-lft-identity 72.5%

          \[\leadsto \frac{\color{blue}{-x}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       5. add-sqr-sqrt 42.3%

          \[\leadsto \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       6. sqrt-unprod 35.8%

          \[\leadsto \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       7. sqr-neg 35.8%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       8. sqrt-unprod 8.2%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       9. add-sqr-sqrt 15.5%

          \[\leadsto \frac{\color{blue}{x}}{\frac{z}{-2} \cdot \left(-t\right)} \]

       10. clear-num 15.5%

           \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{-2}{z}}} \cdot \left(-t\right)} \]

       11. add-sqr-sqrt 8.6%

           \[\leadsto \frac{x}{\frac{1}{\color{blue}{\sqrt{\frac{-2}{z}} \cdot \sqrt{\frac{-2}{z}}}} \cdot \left(-t\right)} \]

       12. sqrt-unprod 30.4%

           \[\leadsto \frac{x}{\frac{1}{\color{blue}{\sqrt{\frac{-2}{z} \cdot \frac{-2}{z}}}} \cdot \left(-t\right)} \]

       13. frac-times 30.4%

           \[\leadsto \frac{x}{\frac{1}{\sqrt{\color{blue}{\frac{-2 \cdot -2}{z \cdot z}}}} \cdot \left(-t\right)} \]

       14. metadata-eval 30.4%

           \[\leadsto \frac{x}{\frac{1}{\sqrt{\frac{\color{blue}{4}}{z \cdot z}}} \cdot \left(-t\right)} \]

       15. metadata-eval 30.4%

           \[\leadsto \frac{x}{\frac{1}{\sqrt{\frac{\color{blue}{2 \cdot 2}}{z \cdot z}}} \cdot \left(-t\right)} \]

       16. frac-times 30.4%

           \[\leadsto \frac{x}{\frac{1}{\sqrt{\color{blue}{\frac{2}{z} \cdot \frac{2}{z}}}} \cdot \left(-t\right)} \]

       17. sqrt-unprod 31.4%

           \[\leadsto \frac{x}{\frac{1}{\color{blue}{\sqrt{\frac{2}{z}} \cdot \sqrt{\frac{2}{z}}}} \cdot \left(-t\right)} \]

       18. add-sqr-sqrt 72.4%

           \[\leadsto \frac{x}{\frac{1}{\color{blue}{\frac{2}{z}}} \cdot \left(-t\right)} \]

       19. clear-num 72.5%

           \[\leadsto \frac{x}{\color{blue}{\frac{z}{2}} \cdot \left(-t\right)} \]

       20. div-inv 72.5%

           \[\leadsto \frac{x}{\color{blue}{\left(z \cdot \frac{1}{2}\right)} \cdot \left(-t\right)} \]

       21. metadata-eval 72.5%

           \[\leadsto \frac{x}{\left(z \cdot \color{blue}{0.5}\right) \cdot \left(-t\right)} \]

   11. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{\frac{x}{\left(z \cdot 0.5\right) \cdot \left(-t\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 72.5%

          \[\leadsto \frac{x}{\color{blue}{\left(-t\right) \cdot \left(z \cdot 0.5\right)}} \]

       2. distribute-lft-neg-out 72.5%

          \[\leadsto \frac{x}{\color{blue}{-t \cdot \left(z \cdot 0.5\right)}} \]

       3. associate-*r* 72.5%

          \[\leadsto \frac{x}{-\color{blue}{\left(t \cdot z\right) \cdot 0.5}} \]

       4. *-commutative 72.5%

          \[\leadsto \frac{x}{-\color{blue}{\left(z \cdot t\right)} \cdot 0.5} \]

       5. distribute-rgt-neg-in 72.5%

          \[\leadsto \frac{x}{\color{blue}{\left(z \cdot t\right) \cdot \left(-0.5\right)}} \]

       6. metadata-eval 72.5%

          \[\leadsto \frac{x}{\left(z \cdot t\right) \cdot \color{blue}{-0.5}} \]

   13. Simplified 72.5%

       \[\leadsto \color{blue}{\frac{x}{\left(z \cdot t\right) \cdot -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-52} \lor \neg \left(y \leq 1.15 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot t\right) \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51} \lor \neg \left(y \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= y -1.2e-51) (not (<= y 6.2e+14)))
    (* x (/ (/ 2.0 z_m) y))
    (* -2.0 (/ x (* z_m t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.2e-51) || !(y <= 6.2e+14)) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * (x / (z_m * t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.2d-51)) .or. (.not. (y <= 6.2d+14))) then
        tmp = x * ((2.0d0 / z_m) / y)
    else
        tmp = (-2.0d0) * (x / (z_m * t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.2e-51) || !(y <= 6.2e+14)) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * (x / (z_m * t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (y <= -1.2e-51) or not (y <= 6.2e+14):
		tmp = x * ((2.0 / z_m) / y)
	else:
		tmp = -2.0 * (x / (z_m * t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((y <= -1.2e-51) || !(y <= 6.2e+14))
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((y <= -1.2e-51) || ~((y <= 6.2e+14)))
		tmp = x * ((2.0 / z_m) / y);
	else
		tmp = -2.0 * (x / (z_m * t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[y, -1.2e-51], N[Not[LessEqual[y, 6.2e+14]], $MachinePrecision]], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e-51 or 6.2e14 < y

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 94.8%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 91.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 91.4%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 91.4%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 94.7%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

   7. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{\frac{2}{y \cdot z}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative 81.8%

         \[\leadsto \frac{2}{\color{blue}{z \cdot y}} \cdot x \]

   9. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{2}{z \cdot y}} \cdot x \]

   10. Taylor expanded in z around 0 81.8%

       \[\leadsto \color{blue}{\frac{2}{y \cdot z}} \cdot x \]
   11. Step-by-step derivation

       1. associate-/l/ 83.0%

          \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y}} \cdot x \]

   12. Simplified 83.0%

       \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y}} \cdot x \]

   if -1.2e-51 < y < 6.2e14

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 92.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto -2 \cdot \frac{x}{\color{blue}{z \cdot t}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{z \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51} \lor \neg \left(y \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{-51} \lor \neg \left(y \leq 6.4 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= y -1.04e-51) (not (<= y 6.4e+14)))
    (* x (/ 2.0 (* y z_m)))
    (* -2.0 (/ x (* z_m t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.04e-51) || !(y <= 6.4e+14)) {
		tmp = x * (2.0 / (y * z_m));
	} else {
		tmp = -2.0 * (x / (z_m * t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.04d-51)) .or. (.not. (y <= 6.4d+14))) then
        tmp = x * (2.0d0 / (y * z_m))
    else
        tmp = (-2.0d0) * (x / (z_m * t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.04e-51) || !(y <= 6.4e+14)) {
		tmp = x * (2.0 / (y * z_m));
	} else {
		tmp = -2.0 * (x / (z_m * t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (y <= -1.04e-51) or not (y <= 6.4e+14):
		tmp = x * (2.0 / (y * z_m))
	else:
		tmp = -2.0 * (x / (z_m * t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((y <= -1.04e-51) || !(y <= 6.4e+14))
		tmp = Float64(x * Float64(2.0 / Float64(y * z_m)));
	else
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((y <= -1.04e-51) || ~((y <= 6.4e+14)))
		tmp = x * (2.0 / (y * z_m));
	else
		tmp = -2.0 * (x / (z_m * t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[y, -1.04e-51], N[Not[LessEqual[y, 6.4e+14]], $MachinePrecision]], N[(x * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.0399999999999999e-51 or 6.4e14 < y

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 94.8%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 91.5%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 91.4%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 91.4%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 94.7%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

   7. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{\frac{2}{y \cdot z}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative 81.8%

         \[\leadsto \frac{2}{\color{blue}{z \cdot y}} \cdot x \]

   9. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{2}{z \cdot y}} \cdot x \]

   if -1.0399999999999999e-51 < y < 6.4e14

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 92.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto -2 \cdot \frac{x}{\color{blue}{z \cdot t}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{z \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{-51} \lor \neg \left(y \leq 6.4 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-9}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (* x 2.0) 1e-9)
    (/ (* x 2.0) (* z_m (- y t)))
    (* (/ 2.0 z_m) (/ x (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-9) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * 2.0d0) <= 1d-9) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (2.0d0 / z_m) * (x / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-9) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (x * 2.0) <= 1e-9:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (2.0 / z_m) * (x / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 1e-9)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((x * 2.0) <= 1e-9)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (2.0 / z_m) * (x / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 1e-9], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 93.3%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   if 1.00000000000000006e-9 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 92.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 93.9%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. times-frac 92.3%

         \[\leadsto \color{blue}{\frac{x}{y - t} \cdot \frac{2}{z}} \]

   6. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{\frac{x}{y - t} \cdot \frac{2}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-9}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-9}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (* x 2.0) 1e-9)
    (* x (/ 2.0 (* z_m (- y t))))
    (* (/ 2.0 z_m) (/ x (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-9) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * 2.0d0) <= 1d-9) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (2.0d0 / z_m) * (x / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-9) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (x * 2.0) <= 1e-9:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (2.0 / z_m) * (x / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 1e-9)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((x * 2.0) <= 1e-9)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (2.0 / z_m) * (x / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 1e-9], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 93.3%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 91.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 91.1%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 93.2%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 93.2%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

   if 1.00000000000000006e-9 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 92.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 93.9%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. times-frac 92.3%

         \[\leadsto \color{blue}{\frac{x}{y - t} \cdot \frac{2}{z}} \]

   6. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{\frac{x}{y - t} \cdot \frac{2}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-9}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e+21)
    (* x (/ 2.0 (* z_m (- y t))))
    (* (/ x z_m) (/ 2.0 (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+21) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2d+21) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+21) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2e+21:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x / z_m) * (2.0 / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2e+21)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2e+21)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (x / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2e+21], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e21

   1. Initial program 93.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 96.0%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-out-- 93.6%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      2. associate-/l* 93.5%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      3. *-commutative 93.5%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      4. distribute-rgt-out-- 95.9%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

   6. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

   if 2e21 < z

   1. Initial program 83.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 83.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. times-frac 92.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   6. Applied egg-rr 92.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.8 \cdot 10^{+21}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.8e+21) (* -2.0 (/ x (* z_m t))) (* -2.0 (/ (/ x z_m) t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.8e+21) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 1.8d+21) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else
        tmp = (-2.0d0) * ((x / z_m) / t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.8e+21) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 1.8e+21:
		tmp = -2.0 * (x / (z_m * t))
	else:
		tmp = -2.0 * ((x / z_m) / t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.8e+21)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 1.8e+21)
		tmp = -2.0 * (x / (z_m * t));
	else
		tmp = -2.0 * ((x / z_m) / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 1.8e+21], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.8e21

   1. Initial program 93.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 96.0%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 56.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative 56.5%

         \[\leadsto -2 \cdot \frac{x}{\color{blue}{z \cdot t}} \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{z \cdot t}} \]

   if 1.8e21 < z

   1. Initial program 83.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. distribute-rgt-out-- 83.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 38.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative 38.9%

         \[\leadsto -2 \cdot \frac{x}{\color{blue}{z \cdot t}} \]

      2. associate-/r* 51.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{\frac{x}{z}}{t}} \]

   7. Simplified 51.8%

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

Alternative 9: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (* z_s (* x (/ 2.0 (* z_m (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (x * (2.0 / (z_m * (y - t))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x * (2.0d0 / (z_m * (y - t))))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (x * (2.0 / (z_m * (y - t))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	return z_s * (x * (2.0 / (z_m * (y - t))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	return Float64(z_s * Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t)))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * (x * (2.0 / (z_m * (y - t))));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
2. Step-by-step derivation

   1. distribute-rgt-out-- 93.4%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

3. Simplified 93.4%

   \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-out-- 91.5%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

   2. associate-/l* 91.4%

      \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

   3. *-commutative 91.4%

      \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

   4. distribute-rgt-out-- 93.4%

      \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

6. Applied egg-rr 93.4%

   \[\leadsto \color{blue}{\frac{2}{z \cdot \left(y - t\right)} \cdot x} \]

7. Final simplification 93.4%

   \[\leadsto x \cdot \frac{2}{z \cdot \left(y - t\right)} \]
8. Add Preprocessing

Alternative 10: 53.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(-2 \cdot \frac{x}{z\_m \cdot t}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t) :precision binary64 (* z_s (* -2.0 (/ x (* z_m t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (-2.0 * (x / (z_m * t)));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * ((-2.0d0) * (x / (z_m * t)))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (-2.0 * (x / (z_m * t)));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	return z_s * (-2.0 * (x / (z_m * t)))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	return Float64(z_s * Float64(-2.0 * Float64(x / Float64(z_m * t))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * (-2.0 * (x / (z_m * t)));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
2. Step-by-step derivation

   1. distribute-rgt-out-- 93.4%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

3. Simplified 93.4%

   \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 53.0%

   \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
6. Step-by-step derivation

   1. *-commutative 53.0%

      \[\leadsto -2 \cdot \frac{x}{\color{blue}{z \cdot t}} \]

7. Simplified 53.0%

   \[\leadsto \color{blue}{-2 \cdot \frac{x}{z \cdot t}} \]
8. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ t_2 := \frac{x \cdot 2}{y \cdot z - t \cdot z}\\ \mathbf{if}\;t\_2 < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (* (- y t) z)) 2.0))
        (t_2 (/ (* x 2.0) (- (* y z) (* t z)))))
   (if (< t_2 -2.559141628295061e-13)
     t_1
     (if (< t_2 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / ((y - t) * z)) * 2.0d0
    t_2 = (x * 2.0d0) / ((y * z) - (t * z))
    if (t_2 < (-2.559141628295061d-13)) then
        tmp = t_1
    else if (t_2 < 1.045027827330126d-269) then
        tmp = ((x / z) * 2.0d0) / (y - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / ((y - t) * z)) * 2.0
	t_2 = (x * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if t_2 < -2.559141628295061e-13:
		tmp = t_1
	elif t_2 < 1.045027827330126e-269:
		tmp = ((x / z) * 2.0) / (y - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0)
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = Float64(Float64(Float64(x / z) * 2.0) / Float64(y - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / ((y - t) * z)) * 2.0;
	t_2 = (x * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = ((x / z) * 2.0) / (y - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -2.559141628295061e-13], t$95$1, If[Less[t$95$2, 1.045027827330126e-269], N[(N[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024128 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (* x 2) (- (* y z) (* t z))) -2559141628295061/10000000000000000000000000000) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 522513913665063/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2))))

  (/ (* x 2.0) (- (* y z) (* t z))))