Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.2% → 97.1%

Time: 10.2s

Alternatives: 12

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: 89.2% 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: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x\_m}{z \cdot 0.5}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - t}}{z \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+14)
    (/ (/ x_m (* z 0.5)) (- y t))
    (/ (/ x_m (- y t)) (* z 0.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+14) {
		tmp = (x_m / (z * 0.5)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) / (z * 0.5);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+14) then
        tmp = (x_m / (z * 0.5d0)) / (y - t)
    else
        tmp = (x_m / (y - t)) / (z * 0.5d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+14) {
		tmp = (x_m / (z * 0.5)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) / (z * 0.5);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+14:
		tmp = (x_m / (z * 0.5)) / (y - t)
	else:
		tmp = (x_m / (y - t)) / (z * 0.5)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+14)
		tmp = Float64(Float64(x_m / Float64(z * 0.5)) / Float64(y - t));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) / Float64(z * 0.5));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+14)
		tmp = (x_m / (z * 0.5)) / (y - t);
	else
		tmp = (x_m / (y - t)) / (z * 0.5);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+14], N[(N[(x$95$m / N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4e14

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. add-sqr-sqrt 34.8%

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

      2. *-commutative 34.8%

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

      3. times-frac 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. frac-times 34.8%

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

      2. add-sqr-sqrt 94.0%

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

      3. frac-times 91.6%

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

      4. *-commutative 91.6%

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

      5. clear-num 91.6%

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

      6. frac-times 94.0%

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

      7. *-un-lft-identity 94.0%

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

      8. div-inv 94.0%

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

      9. metadata-eval 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/r* 93.3%

         \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]

   10. Simplified 93.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]

   if 4e14 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 87.6%

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

   3. Simplified 87.6%

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

      1. add-sqr-sqrt 87.1%

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

      2. *-commutative 87.1%

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

      3. times-frac 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. frac-times 87.1%

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

      2. add-sqr-sqrt 87.6%

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

      3. frac-times 96.6%

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

      4. clear-num 96.6%

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

      5. un-div-inv 96.7%

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

      6. div-inv 96.7%

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

      7. metadata-eval 96.7%

         \[\leadsto \frac{\frac{x}{y - t}}{z \cdot \color{blue}{0.5}} \]

   8. Applied egg-rr 96.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{\frac{x\_m}{z}}{t}\\ t_2 := x\_m \cdot \frac{2}{z \cdot y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-223}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;y \leq 1450000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* -2.0 (/ (/ x_m z) t))) (t_2 (* x_m (/ 2.0 (* z y)))))
   (*
    x_s
    (if (<= y -5.2e-51)
      t_2
      (if (<= y -8.6e-75)
        t_1
        (if (<= y -1.4e-95)
          (* (/ 2.0 z) (/ x_m y))
          (if (<= y -1e-223)
            (* -2.0 (/ x_m (* z t)))
            (if (<= y 1450000000.0) t_1 t_2))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -2.0 * ((x_m / z) / t);
	double t_2 = x_m * (2.0 / (z * y));
	double tmp;
	if (y <= -5.2e-51) {
		tmp = t_2;
	} else if (y <= -8.6e-75) {
		tmp = t_1;
	} else if (y <= -1.4e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1e-223) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 1450000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 = (-2.0d0) * ((x_m / z) / t)
    t_2 = x_m * (2.0d0 / (z * y))
    if (y <= (-5.2d-51)) then
        tmp = t_2
    else if (y <= (-8.6d-75)) then
        tmp = t_1
    else if (y <= (-1.4d-95)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= (-1d-223)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (y <= 1450000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -2.0 * ((x_m / z) / t);
	double t_2 = x_m * (2.0 / (z * y));
	double tmp;
	if (y <= -5.2e-51) {
		tmp = t_2;
	} else if (y <= -8.6e-75) {
		tmp = t_1;
	} else if (y <= -1.4e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1e-223) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 1450000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = -2.0 * ((x_m / z) / t)
	t_2 = x_m * (2.0 / (z * y))
	tmp = 0
	if y <= -5.2e-51:
		tmp = t_2
	elif y <= -8.6e-75:
		tmp = t_1
	elif y <= -1.4e-95:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= -1e-223:
		tmp = -2.0 * (x_m / (z * t))
	elif y <= 1450000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(-2.0 * Float64(Float64(x_m / z) / t))
	t_2 = Float64(x_m * Float64(2.0 / Float64(z * y)))
	tmp = 0.0
	if (y <= -5.2e-51)
		tmp = t_2;
	elseif (y <= -8.6e-75)
		tmp = t_1;
	elseif (y <= -1.4e-95)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= -1e-223)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (y <= 1450000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = -2.0 * ((x_m / z) / t);
	t_2 = x_m * (2.0 / (z * y));
	tmp = 0.0;
	if (y <= -5.2e-51)
		tmp = t_2;
	elseif (y <= -8.6e-75)
		tmp = t_1;
	elseif (y <= -1.4e-95)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= -1e-223)
		tmp = -2.0 * (x_m / (z * t));
	elseif (y <= 1450000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5.2e-51], t$95$2, If[LessEqual[y, -8.6e-75], t$95$1, If[LessEqual[y, -1.4e-95], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1e-223], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1450000000.0], t$95$1, t$95$2]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.2e-51 or 1.45e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. associate-/l* 84.2%

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

      2. *-commutative 84.2%

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

   9. Applied egg-rr 84.2%

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

   if -5.2e-51 < y < -8.5999999999999998e-75 or -9.9999999999999997e-224 < y < 1.45e9

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y around 0 81.7%

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

      1. *-commutative 81.7%

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

   7. Simplified 81.7%

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

      1. associate-/r* 85.5%

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

      2. div-inv 85.4%

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

   9. Applied egg-rr 85.4%

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

       1. associate-*r/ 85.5%

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

       2. *-rgt-identity 85.5%

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

   11. Simplified 85.5%

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

   if -8.5999999999999998e-75 < y < -1.4e-95

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 87.4%

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

   if -1.4e-95 < y < -9.9999999999999997e-224

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 82.7%

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

      1. *-commutative 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-223}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq 1450000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{2}{z \cdot y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;y \leq 3700000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ 2.0 (* z y)))))
   (*
    x_s
    (if (<= y -5e-51)
      t_1
      (if (<= y -5.8e-75)
        (/ -2.0 (* t (/ z x_m)))
        (if (<= y -1.45e-95)
          (* (/ 2.0 z) (/ x_m y))
          (if (<= y -1.7e-220)
            (* -2.0 (/ x_m (* z t)))
            (if (<= y 3700000000.0) (* -2.0 (/ (/ x_m z) t)) t_1))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (z * y));
	double tmp;
	if (y <= -5e-51) {
		tmp = t_1;
	} else if (y <= -5.8e-75) {
		tmp = -2.0 / (t * (z / x_m));
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1.7e-220) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 3700000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (2.0d0 / (z * y))
    if (y <= (-5d-51)) then
        tmp = t_1
    else if (y <= (-5.8d-75)) then
        tmp = (-2.0d0) / (t * (z / x_m))
    else if (y <= (-1.45d-95)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= (-1.7d-220)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (y <= 3700000000.0d0) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (z * y));
	double tmp;
	if (y <= -5e-51) {
		tmp = t_1;
	} else if (y <= -5.8e-75) {
		tmp = -2.0 / (t * (z / x_m));
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1.7e-220) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 3700000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (2.0 / (z * y))
	tmp = 0
	if y <= -5e-51:
		tmp = t_1
	elif y <= -5.8e-75:
		tmp = -2.0 / (t * (z / x_m))
	elif y <= -1.45e-95:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= -1.7e-220:
		tmp = -2.0 * (x_m / (z * t))
	elif y <= 3700000000.0:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(2.0 / Float64(z * y)))
	tmp = 0.0
	if (y <= -5e-51)
		tmp = t_1;
	elseif (y <= -5.8e-75)
		tmp = Float64(-2.0 / Float64(t * Float64(z / x_m)));
	elseif (y <= -1.45e-95)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= -1.7e-220)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (y <= 3700000000.0)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (2.0 / (z * y));
	tmp = 0.0;
	if (y <= -5e-51)
		tmp = t_1;
	elseif (y <= -5.8e-75)
		tmp = -2.0 / (t * (z / x_m));
	elseif (y <= -1.45e-95)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= -1.7e-220)
		tmp = -2.0 * (x_m / (z * t));
	elseif (y <= 3700000000.0)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5e-51], t$95$1, If[LessEqual[y, -5.8e-75], N[(-2.0 / N[(t * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.45e-95], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e-220], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3700000000.0], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -5.00000000000000004e-51 or 3.7e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. associate-/l* 84.2%

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

      2. *-commutative 84.2%

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

   9. Applied egg-rr 84.2%

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

   if -5.00000000000000004e-51 < y < -5.8000000000000003e-75

   1. Initial program 78.5%

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

      1. distribute-rgt-out-- 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in y around 0 67.7%

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

      1. *-commutative 67.7%

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

   7. Simplified 67.7%

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

      1. clear-num 67.8%

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

      2. un-div-inv 67.8%

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

      3. *-commutative 67.8%

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

   9. Applied egg-rr 67.8%

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

       1. associate-/l* 78.3%

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

   11. Simplified 78.3%

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

   if -5.8000000000000003e-75 < y < -1.45000000000000001e-95

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 87.4%

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

   if -1.45000000000000001e-95 < y < -1.69999999999999997e-220

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 82.7%

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

      1. *-commutative 82.7%

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

   7. Simplified 82.7%

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

   if -1.69999999999999997e-220 < y < 3.7e9

   1. Initial program 90.4%

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

      1. distribute-rgt-out-- 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

      1. associate-/r* 86.5%

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

      2. div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

       1. associate-*r/ 86.5%

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

       2. *-rgt-identity 86.5%

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

   11. Simplified 86.5%

       \[\leadsto -2 \cdot \color{blue}{\frac{\frac{x}{z}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq 3700000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{z \cdot y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq -1.16 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;y \leq 2200000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m 2.0) (* z y))))
   (*
    x_s
    (if (<= y -2.4e-52)
      t_1
      (if (<= y -1.8e-74)
        (/ -2.0 (* t (/ z x_m)))
        (if (<= y -1.45e-95)
          (* (/ 2.0 z) (/ x_m y))
          (if (<= y -1.16e-220)
            (* -2.0 (/ x_m (* z t)))
            (if (<= y 2200000000.0) (* -2.0 (/ (/ x_m z) t)) t_1))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -2.4e-52) {
		tmp = t_1;
	} else if (y <= -1.8e-74) {
		tmp = -2.0 / (t * (z / x_m));
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1.16e-220) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 2200000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / (z * y)
    if (y <= (-2.4d-52)) then
        tmp = t_1
    else if (y <= (-1.8d-74)) then
        tmp = (-2.0d0) / (t * (z / x_m))
    else if (y <= (-1.45d-95)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= (-1.16d-220)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (y <= 2200000000.0d0) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -2.4e-52) {
		tmp = t_1;
	} else if (y <= -1.8e-74) {
		tmp = -2.0 / (t * (z / x_m));
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= -1.16e-220) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (y <= 2200000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * 2.0) / (z * y)
	tmp = 0
	if y <= -2.4e-52:
		tmp = t_1
	elif y <= -1.8e-74:
		tmp = -2.0 / (t * (z / x_m))
	elif y <= -1.45e-95:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= -1.16e-220:
		tmp = -2.0 * (x_m / (z * t))
	elif y <= 2200000000.0:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(z * y))
	tmp = 0.0
	if (y <= -2.4e-52)
		tmp = t_1;
	elseif (y <= -1.8e-74)
		tmp = Float64(-2.0 / Float64(t * Float64(z / x_m)));
	elseif (y <= -1.45e-95)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= -1.16e-220)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (y <= 2200000000.0)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * 2.0) / (z * y);
	tmp = 0.0;
	if (y <= -2.4e-52)
		tmp = t_1;
	elseif (y <= -1.8e-74)
		tmp = -2.0 / (t * (z / x_m));
	elseif (y <= -1.45e-95)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= -1.16e-220)
		tmp = -2.0 * (x_m / (z * t));
	elseif (y <= 2200000000.0)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.4e-52], t$95$1, If[LessEqual[y, -1.8e-74], N[(-2.0 / N[(t * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.45e-95], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.16e-220], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2200000000.0], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -2.4000000000000002e-52 or 2.2e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

   if -2.4000000000000002e-52 < y < -1.8000000000000001e-74

   1. Initial program 78.5%

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

      1. distribute-rgt-out-- 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in y around 0 67.7%

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

      1. *-commutative 67.7%

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

   7. Simplified 67.7%

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

      1. clear-num 67.8%

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

      2. un-div-inv 67.8%

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

      3. *-commutative 67.8%

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

   9. Applied egg-rr 67.8%

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

       1. associate-/l* 78.3%

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

   11. Simplified 78.3%

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

   if -1.8000000000000001e-74 < y < -1.45000000000000001e-95

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 87.4%

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

   if -1.45000000000000001e-95 < y < -1.15999999999999998e-220

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 82.7%

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

      1. *-commutative 82.7%

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

   7. Simplified 82.7%

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

   if -1.15999999999999998e-220 < y < 2.2e9

   1. Initial program 90.4%

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

      1. distribute-rgt-out-- 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

      1. associate-/r* 86.5%

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

      2. div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

       1. associate-*r/ 86.5%

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

       2. *-rgt-identity 86.5%

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

   11. Simplified 86.5%

       \[\leadsto -2 \cdot \color{blue}{\frac{\frac{x}{z}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.16 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq 2200000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{z \cdot y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{2}{-z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 1450000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m 2.0) (* z y))))
   (*
    x_s
    (if (<= y -3.15e-52)
      t_1
      (if (<= y -2.6e-74)
        (* (/ x_m t) (/ 2.0 (- z)))
        (if (<= y -1.45e-95)
          (* (/ 2.0 z) (/ x_m y))
          (if (<= y 1450000000.0) (* -2.0 (/ (/ x_m z) t)) t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -3.15e-52) {
		tmp = t_1;
	} else if (y <= -2.6e-74) {
		tmp = (x_m / t) * (2.0 / -z);
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 1450000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / (z * y)
    if (y <= (-3.15d-52)) then
        tmp = t_1
    else if (y <= (-2.6d-74)) then
        tmp = (x_m / t) * (2.0d0 / -z)
    else if (y <= (-1.45d-95)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= 1450000000.0d0) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -3.15e-52) {
		tmp = t_1;
	} else if (y <= -2.6e-74) {
		tmp = (x_m / t) * (2.0 / -z);
	} else if (y <= -1.45e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 1450000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * 2.0) / (z * y)
	tmp = 0
	if y <= -3.15e-52:
		tmp = t_1
	elif y <= -2.6e-74:
		tmp = (x_m / t) * (2.0 / -z)
	elif y <= -1.45e-95:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= 1450000000.0:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(z * y))
	tmp = 0.0
	if (y <= -3.15e-52)
		tmp = t_1;
	elseif (y <= -2.6e-74)
		tmp = Float64(Float64(x_m / t) * Float64(2.0 / Float64(-z)));
	elseif (y <= -1.45e-95)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= 1450000000.0)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * 2.0) / (z * y);
	tmp = 0.0;
	if (y <= -3.15e-52)
		tmp = t_1;
	elseif (y <= -2.6e-74)
		tmp = (x_m / t) * (2.0 / -z);
	elseif (y <= -1.45e-95)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= 1450000000.0)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -3.15e-52], t$95$1, If[LessEqual[y, -2.6e-74], N[(N[(x$95$m / t), $MachinePrecision] * N[(2.0 / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.45e-95], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1450000000.0], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1500000000000002e-52 or 1.45e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

   if -3.1500000000000002e-52 < y < -2.6000000000000001e-74

   1. Initial program 78.5%

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

      1. distribute-rgt-out-- 78.5%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. times-frac 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 88.8%

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

      1. associate-*r/ 88.8%

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

      2. neg-mul-1 88.8%

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

   9. Simplified 88.8%

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

   if -2.6000000000000001e-74 < y < -1.45000000000000001e-95

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 87.4%

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

   if -1.45000000000000001e-95 < y < 1.45e9

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in y around 0 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

      1. associate-/r* 85.0%

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

      2. div-inv 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. associate-*r/ 85.0%

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

       2. *-rgt-identity 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-52}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{2}{-z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1450000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{z \cdot y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{-x\_m}{t}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 2400000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m 2.0) (* z y))))
   (*
    x_s
    (if (<= y -5.1e-51)
      t_1
      (if (<= y -1.2e-73)
        (/ (/ (- x_m) t) (* z 0.5))
        (if (<= y -1.35e-95)
          (* (/ 2.0 z) (/ x_m y))
          (if (<= y 2400000000.0) (* -2.0 (/ (/ x_m z) t)) t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -5.1e-51) {
		tmp = t_1;
	} else if (y <= -1.2e-73) {
		tmp = (-x_m / t) / (z * 0.5);
	} else if (y <= -1.35e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 2400000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / (z * y)
    if (y <= (-5.1d-51)) then
        tmp = t_1
    else if (y <= (-1.2d-73)) then
        tmp = (-x_m / t) / (z * 0.5d0)
    else if (y <= (-1.35d-95)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= 2400000000.0d0) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * 2.0) / (z * y);
	double tmp;
	if (y <= -5.1e-51) {
		tmp = t_1;
	} else if (y <= -1.2e-73) {
		tmp = (-x_m / t) / (z * 0.5);
	} else if (y <= -1.35e-95) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 2400000000.0) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * 2.0) / (z * y)
	tmp = 0
	if y <= -5.1e-51:
		tmp = t_1
	elif y <= -1.2e-73:
		tmp = (-x_m / t) / (z * 0.5)
	elif y <= -1.35e-95:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= 2400000000.0:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(z * y))
	tmp = 0.0
	if (y <= -5.1e-51)
		tmp = t_1;
	elseif (y <= -1.2e-73)
		tmp = Float64(Float64(Float64(-x_m) / t) / Float64(z * 0.5));
	elseif (y <= -1.35e-95)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= 2400000000.0)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * 2.0) / (z * y);
	tmp = 0.0;
	if (y <= -5.1e-51)
		tmp = t_1;
	elseif (y <= -1.2e-73)
		tmp = (-x_m / t) / (z * 0.5);
	elseif (y <= -1.35e-95)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= 2400000000.0)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5.1e-51], t$95$1, If[LessEqual[y, -1.2e-73], N[(N[((-x$95$m) / t), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.35e-95], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2400000000.0], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0999999999999997e-51 or 2.4e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

   if -5.0999999999999997e-51 < y < -1.20000000000000003e-73

   1. Initial program 78.5%

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

      1. distribute-rgt-out-- 78.5%

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

   3. Simplified 78.5%

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

      1. add-sqr-sqrt 44.1%

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

      2. *-commutative 44.1%

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

      3. times-frac 44.3%

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

   6. Applied egg-rr 44.3%

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

      1. frac-times 44.1%

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

      2. add-sqr-sqrt 78.5%

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

      3. frac-times 99.7%

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

      4. clear-num 99.7%

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

      5. un-div-inv 99.7%

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

      6. div-inv 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto \frac{\frac{x}{y - t}}{z \cdot \color{blue}{0.5}} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y - t}}{z \cdot 0.5}} \]

   9. Taylor expanded in y around 0 88.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{x}{t}}}{z \cdot 0.5} \]
   10. Step-by-step derivation

       1. associate-*r/ 88.8%

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

       2. neg-mul-1 88.8%

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

   11. Simplified 88.9%

       \[\leadsto \frac{\color{blue}{\frac{-x}{t}}}{z \cdot 0.5} \]

   if -1.20000000000000003e-73 < y < -1.35e-95

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 87.4%

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

   if -1.35e-95 < y < 2.4e9

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in y around 0 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

      1. associate-/r* 85.0%

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

      2. div-inv 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. associate-*r/ 85.0%

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

       2. *-rgt-identity 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2400000000:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-51} \lor \neg \left(y \leq 1900000000\right):\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -4.7e-51) (not (<= y 1900000000.0)))
    (* x_m (/ 2.0 (* z y)))
    (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -4.7e-51) || !(y <= 1900000000.0)) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4.7d-51)) .or. (.not. (y <= 1900000000.0d0))) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -4.7e-51) || !(y <= 1900000000.0)) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -4.7e-51) or not (y <= 1900000000.0):
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -4.7e-51) || !(y <= 1900000000.0))
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -4.7e-51) || ~((y <= 1900000000.0)))
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -4.7e-51], N[Not[LessEqual[y, 1900000000.0]], $MachinePrecision]], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.6999999999999997e-51 or 1.9e9 < y

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. associate-/l* 84.2%

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

      2. *-commutative 84.2%

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

   9. Applied egg-rr 84.2%

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

   if -4.6999999999999997e-51 < y < 1.9e9

   1. Initial program 88.5%

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

      1. distribute-rgt-out-- 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in y around 0 78.0%

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

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

      1. associate-/r* 81.0%

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

      2. div-inv 80.9%

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

   9. Applied egg-rr 80.9%

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

       1. associate-*r/ 81.0%

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

       2. *-rgt-identity 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-51} \lor \neg \left(y \leq 1900000000\right):\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e-41)
    (* 2.0 (/ (/ x_m z) (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-41) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-41) then
        tmp = 2.0d0 * ((x_m / z) / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-41) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-41:
		tmp = 2.0 * ((x_m / z) / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-41)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-41)
		tmp = 2.0 * ((x_m / z) / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-41], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.9999999999999996e-41

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in x around 0 93.7%

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

      1. associate-/r* 92.9%

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

   7. Simplified 92.9%

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

   if 4.9999999999999996e-41 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 89.2%

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

   3. Simplified 89.2%

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

      1. *-commutative 89.2%

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

      2. times-frac 97.0%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{x\_m}{z \cdot 0.5}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e-41)
    (/ (/ x_m (* z 0.5)) (- y t))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-41) {
		tmp = (x_m / (z * 0.5)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-41) then
        tmp = (x_m / (z * 0.5d0)) / (y - t)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-41) {
		tmp = (x_m / (z * 0.5)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-41:
		tmp = (x_m / (z * 0.5)) / (y - t)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-41)
		tmp = Float64(Float64(x_m / Float64(z * 0.5)) / Float64(y - t));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-41)
		tmp = (x_m / (z * 0.5)) / (y - t);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-41], N[(N[(x$95$m / N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.9999999999999996e-41

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 31.1%

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

   6. Applied egg-rr 31.1%

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

      1. frac-times 31.7%

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

      2. add-sqr-sqrt 93.7%

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

      3. frac-times 91.2%

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

      4. *-commutative 91.2%

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

      5. clear-num 91.2%

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

      6. frac-times 93.7%

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

      7. *-un-lft-identity 93.7%

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

      8. div-inv 93.7%

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

      9. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. associate-/r* 93.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]

   10. Simplified 93.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]

   if 4.9999999999999996e-41 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 89.2%

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

   3. Simplified 89.2%

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

      1. *-commutative 89.2%

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

      2. times-frac 97.0%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z 2.5e-24) (* -2.0 (/ x_m (* z t))) (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e-24) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2.5d-24) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e-24) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 2.5e-24:
		tmp = -2.0 * (x_m / (z * t))
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 2.5e-24)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 2.5e-24)
		tmp = -2.0 * (x_m / (z * t));
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 2.5e-24], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.4999999999999999e-24

   1. Initial program 91.2%

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

      1. distribute-rgt-out-- 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around 0 47.9%

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

      1. *-commutative 47.9%

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

   7. Simplified 47.9%

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

   if 2.4999999999999999e-24 < z

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in y around 0 46.2%

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

      1. *-commutative 46.2%

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

   7. Simplified 46.2%

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

      1. associate-/r* 64.2%

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

      2. div-inv 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. associate-*r/ 64.2%

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

       2. *-rgt-identity 64.2%

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

   11. Simplified 64.2%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. distribute-rgt-out-- 92.5%

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

3. Simplified 92.5%

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

5. Taylor expanded in x around 0 92.5%

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

   1. associate-/r* 92.2%

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

7. Simplified 92.2%

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

8. Final simplification 92.2%

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

Alternative 12: 53.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. distribute-rgt-out-- 92.5%

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

3. Simplified 92.5%

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

5. Taylor expanded in y around 0 47.6%

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

   1. *-commutative 47.6%

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

7. Simplified 47.6%

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

8. Final simplification 47.6%

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

Developer target: 96.9% 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 2024078 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

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