Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 96.4%

Time: 11.1s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x\_m \cdot 2}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (sqrt (* x_m 2.0))))
   (*
    z_s
    (*
     x_s
     (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -1e-312)
       (/ (* x_m 2.0) (* z_m (- y t)))
       (* (/ t_1 (- y t)) (/ t_1 z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x_m * 2.0d0))
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-1d-312)) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (t_1 / (y - t)) * (t_1 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = Math.sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = math.sqrt((x_m * 2.0))
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (t_1 / (y - t)) * (t_1 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = sqrt(Float64(x_m * 2.0))
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -1e-312)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(t_1 / Float64(y - t)) * Float64(t_1 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = sqrt((x_m * 2.0));
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[Sqrt[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-312], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -9.9999999999847e-313

   1. Initial program 96.3%

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

      1. distribute-rgt-out-- 97.4%

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

   3. Simplified 97.4%

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

   if -9.9999999999847e-313 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 86.6%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. add-sqr-sqrt 44.5%

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

      2. *-commutative 44.5%

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

      3. times-frac 45.8%

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

   6. Applied egg-rr 45.8%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x \cdot 2}}{y - t} \cdot \frac{\sqrt{x \cdot 2}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -1e-312)
     (/ (* x_m 2.0) (* z_m (- y t)))
     (* (/ x_m (- y t)) (/ 2.0 z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-1d-312)) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -1e-312)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-312)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-312], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -9.9999999999847e-313

   1. Initial program 96.3%

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

      1. distribute-rgt-out-- 97.4%

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

   3. Simplified 97.4%

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

   if -9.9999999999847e-313 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 86.6%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 93.4%

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

   6. Applied egg-rr 93.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z\_m}{x\_m}}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -3.5e-83)
     (/ (/ -2.0 t) (/ z_m x_m))
     (if (<= t 9.4e-23)
       (/ (* 2.0 (/ x_m y)) z_m)
       (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.5e-83) {
		tmp = (-2.0 / t) / (z_m / x_m);
	} else if (t <= 9.4e-23) {
		tmp = (2.0 * (x_m / y)) / z_m;
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.5d-83)) then
        tmp = ((-2.0d0) / t) / (z_m / x_m)
    else if (t <= 9.4d-23) then
        tmp = (2.0d0 * (x_m / y)) / z_m
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.5e-83) {
		tmp = (-2.0 / t) / (z_m / x_m);
	} else if (t <= 9.4e-23) {
		tmp = (2.0 * (x_m / y)) / z_m;
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -3.5e-83:
		tmp = (-2.0 / t) / (z_m / x_m)
	elif t <= 9.4e-23:
		tmp = (2.0 * (x_m / y)) / z_m
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -3.5e-83)
		tmp = Float64(Float64(-2.0 / t) / Float64(z_m / x_m));
	elseif (t <= 9.4e-23)
		tmp = Float64(Float64(2.0 * Float64(x_m / y)) / z_m);
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -3.5e-83)
		tmp = (-2.0 / t) / (z_m / x_m);
	elseif (t <= 9.4e-23)
		tmp = (2.0 * (x_m / y)) / z_m;
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -3.5e-83], N[(N[(-2.0 / t), $MachinePrecision] / N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.4e-23], N[(N[(2.0 * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.5000000000000003e-83

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

      1. times-frac 94.8%

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

   6. Applied egg-rr 94.8%

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

   7. Taylor expanded in y around 0 77.8%

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

      1. *-commutative 77.8%

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

      2. clear-num 77.8%

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

      3. div-inv 78.7%

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

   9. Applied egg-rr 78.7%

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

   if -3.5000000000000003e-83 < t < 9.4000000000000001e-23

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around inf 73.5%

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

      1. associate-/r* 78.9%

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

      2. associate-*r/ 78.9%

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

   7. Simplified 78.9%

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

   if 9.4000000000000001e-23 < t

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num 79.1%

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

      2. un-div-inv 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 75.3%

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

   9. Applied egg-rr 75.3%

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

       1. associate-/r* 76.3%

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

       2. associate-/r/ 80.0%

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

   11. Applied egg-rr 80.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-82}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -1.35e-82)
     (/ -2.0 (* t (/ z_m x_m)))
     (if (<= t 1.5e-22)
       (/ (* 2.0 (/ x_m y)) z_m)
       (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.35e-82) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else if (t <= 1.5e-22) {
		tmp = (2.0 * (x_m / y)) / z_m;
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.35d-82)) then
        tmp = (-2.0d0) / (t * (z_m / x_m))
    else if (t <= 1.5d-22) then
        tmp = (2.0d0 * (x_m / y)) / z_m
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.35e-82) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else if (t <= 1.5e-22) {
		tmp = (2.0 * (x_m / y)) / z_m;
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -1.35e-82:
		tmp = -2.0 / (t * (z_m / x_m))
	elif t <= 1.5e-22:
		tmp = (2.0 * (x_m / y)) / z_m
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -1.35e-82)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x_m)));
	elseif (t <= 1.5e-22)
		tmp = Float64(Float64(2.0 * Float64(x_m / y)) / z_m);
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.35e-82)
		tmp = -2.0 / (t * (z_m / x_m));
	elseif (t <= 1.5e-22)
		tmp = (2.0 * (x_m / y)) / z_m;
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -1.35e-82], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e-22], N[(N[(2.0 * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.3500000000000001e-82

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 75.1%

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

      1. *-commutative 75.1%

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

   7. Simplified 75.1%

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

      1. clear-num 75.0%

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

      2. un-div-inv 75.0%

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

      3. *-commutative 75.0%

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

      4. associate-/l* 78.7%

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

   9. Applied egg-rr 78.7%

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

   if -1.3500000000000001e-82 < t < 1.5e-22

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around inf 73.5%

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

      1. associate-/r* 78.9%

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

      2. associate-*r/ 78.9%

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

   7. Simplified 78.9%

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

   if 1.5e-22 < t

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num 79.1%

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

      2. un-div-inv 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 75.3%

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

   9. Applied egg-rr 75.3%

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

       1. associate-/r* 76.3%

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

       2. associate-/r/ 80.0%

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

   11. Applied egg-rr 80.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-82}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-22}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -6.4e-83)
     (/ -2.0 (* t (/ z_m x_m)))
     (if (<= t 2.15e-22)
       (* (/ x_m z_m) (/ 2.0 y))
       (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -6.4e-83) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else if (t <= 2.15e-22) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-6.4d-83)) then
        tmp = (-2.0d0) / (t * (z_m / x_m))
    else if (t <= 2.15d-22) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -6.4e-83) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else if (t <= 2.15e-22) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -6.4e-83:
		tmp = -2.0 / (t * (z_m / x_m))
	elif t <= 2.15e-22:
		tmp = (x_m / z_m) * (2.0 / y)
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -6.4e-83)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x_m)));
	elseif (t <= 2.15e-22)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -6.4e-83)
		tmp = -2.0 / (t * (z_m / x_m));
	elseif (t <= 2.15e-22)
		tmp = (x_m / z_m) * (2.0 / y);
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -6.4e-83], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e-22], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.4000000000000002e-83

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 75.1%

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

      1. *-commutative 75.1%

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

   7. Simplified 75.1%

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

      1. clear-num 75.0%

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

      2. un-div-inv 75.0%

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

      3. *-commutative 75.0%

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

      4. associate-/l* 78.7%

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

   9. Applied egg-rr 78.7%

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

   if -6.4000000000000002e-83 < t < 2.15000000000000019e-22

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

      1. times-frac 91.0%

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

   6. Applied egg-rr 91.0%

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

   7. Taylor expanded in y around inf 77.9%

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

   if 2.15000000000000019e-22 < t

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num 79.1%

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

      2. un-div-inv 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 75.3%

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

   9. Applied egg-rr 75.3%

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

       1. associate-/r* 76.3%

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

       2. associate-/r/ 80.0%

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

   11. Applied egg-rr 80.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-83}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -5e-83)
     (* -2.0 (/ (/ x_m z_m) t))
     (if (<= t 6.2e-19)
       (* (/ x_m z_m) (/ 2.0 y))
       (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -5e-83) {
		tmp = -2.0 * ((x_m / z_m) / t);
	} else if (t <= 6.2e-19) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-83)) then
        tmp = (-2.0d0) * ((x_m / z_m) / t)
    else if (t <= 6.2d-19) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -5e-83) {
		tmp = -2.0 * ((x_m / z_m) / t);
	} else if (t <= 6.2e-19) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -5e-83:
		tmp = -2.0 * ((x_m / z_m) / t)
	elif t <= 6.2e-19:
		tmp = (x_m / z_m) * (2.0 / y)
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -5e-83)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z_m) / t));
	elseif (t <= 6.2e-19)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -5e-83)
		tmp = -2.0 * ((x_m / z_m) / t);
	elseif (t <= 6.2e-19)
		tmp = (x_m / z_m) * (2.0 / y);
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -5e-83], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-19], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5e-83

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-/r* 78.0%

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

   7. Simplified 78.0%

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

   if -5e-83 < t < 6.1999999999999998e-19

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

      1. times-frac 91.0%

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

   6. Applied egg-rr 91.0%

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

   7. Taylor expanded in y around inf 77.9%

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

   if 6.1999999999999998e-19 < t

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num 79.1%

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

      2. un-div-inv 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 75.3%

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

   9. Applied egg-rr 75.3%

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

       1. associate-/r* 76.3%

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

       2. associate-/r/ 80.0%

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

   11. Applied egg-rr 80.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-83}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-83}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -9e-83)
     (* -2.0 (/ (/ x_m z_m) t))
     (if (<= t 2.8e+42)
       (* (/ x_m z_m) (/ 2.0 y))
       (* -2.0 (/ x_m (* z_m t))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -9e-83) {
		tmp = -2.0 * ((x_m / z_m) / t);
	} else if (t <= 2.8e+42) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d-83)) then
        tmp = (-2.0d0) * ((x_m / z_m) / t)
    else if (t <= 2.8d+42) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else
        tmp = (-2.0d0) * (x_m / (z_m * t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -9e-83) {
		tmp = -2.0 * ((x_m / z_m) / t);
	} else if (t <= 2.8e+42) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -9e-83:
		tmp = -2.0 * ((x_m / z_m) / t)
	elif t <= 2.8e+42:
		tmp = (x_m / z_m) * (2.0 / y)
	else:
		tmp = -2.0 * (x_m / (z_m * t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -9e-83)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z_m) / t));
	elseif (t <= 2.8e+42)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -9e-83)
		tmp = -2.0 * ((x_m / z_m) / t);
	elseif (t <= 2.8e+42)
		tmp = (x_m / z_m) * (2.0 / y);
	else
		tmp = -2.0 * (x_m / (z_m * t));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -9e-83], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+42], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -8.99999999999999995e-83

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-/r* 78.0%

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

   7. Simplified 78.0%

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

   if -8.99999999999999995e-83 < t < 2.7999999999999999e42

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

      1. times-frac 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around inf 76.0%

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

   if 2.7999999999999999e42 < t

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in y around 0 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

Alternative 8: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;x\_m \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 2e+141)
     (* x_m (/ 2.0 (* z_m (- y t))))
     (* (/ x_m (- y t)) (/ 2.0 z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e+141) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 2d+141) then
        tmp = x_m * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e+141) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 2e+141:
		tmp = x_m * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 2e+141)
		tmp = Float64(x_m * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 2e+141)
		tmp = x_m * (2.0 / (z_m * (y - t)));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 2e+141], N[(x$95$m * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2.00000000000000003e141

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

      1. distribute-rgt-out-- 91.5%

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

      2. associate-/l* 90.9%

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

      3. *-commutative 90.9%

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

      4. distribute-rgt-out-- 93.8%

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

   6. Applied egg-rr 93.8%

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

   if 2.00000000000000003e141 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 81.3%

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

      1. distribute-rgt-out-- 84.1%

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

   3. Simplified 84.1%

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

      1. *-commutative 84.1%

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

      2. times-frac 94.4%

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

   6. Applied egg-rr 94.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;x\_m \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 6.5e+83)
     (* x_m (/ 2.0 (* z_m (- y t))))
     (* (/ x_m z_m) (/ 2.0 (- y t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 6.5e+83) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 6.5d+83) then
        tmp = x_m * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 6.5e+83) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 6.5e+83:
		tmp = x_m * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 6.5e+83)
		tmp = Float64(x_m * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 6.5e+83)
		tmp = x_m * (2.0 / (z_m * (y - t)));
	else
		tmp = (x_m / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.5e+83], N[(x$95$m * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.5000000000000003e83

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

      1. distribute-rgt-out-- 92.4%

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

      2. associate-/l* 91.8%

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

      3. *-commutative 91.8%

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

      4. distribute-rgt-out-- 93.3%

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

   6. Applied egg-rr 93.3%

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

   if 6.5000000000000003e83 < z

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

      1. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{+82}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 3.2e+82)
     (* -2.0 (/ x_m (* z_m t)))
     (* (/ x_m z_m) (/ -2.0 t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.2e+82) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (x_m / z_m) * (-2.0 / t);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 3.2d+82) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (x_m / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.2e+82) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (x_m / z_m) * (-2.0 / t);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 3.2e+82:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = (x_m / z_m) * (-2.0 / t)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.2e+82)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 3.2e+82)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = (x_m / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.2e+82], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999975e82

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in y around 0 62.5%

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

      1. *-commutative 62.5%

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

   7. Simplified 62.5%

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

   if 3.19999999999999975e82 < z

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

      1. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 68.6%

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

Alternative 11: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 3.5e+82)
     (* -2.0 (/ x_m (* z_m t)))
     (* -2.0 (/ (/ x_m z_m) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+82) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / z_m) / t);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 3.5d+82) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (-2.0d0) * ((x_m / z_m) / t)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+82) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / z_m) / t);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 3.5e+82:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = -2.0 * ((x_m / z_m) / t)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.5e+82)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z_m) / t));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 3.5e+82)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = -2.0 * ((x_m / z_m) / t);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.5e+82], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5e82

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in y around 0 62.5%

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

      1. *-commutative 62.5%

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

   7. Simplified 62.5%

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

   if 3.5e82 < z

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in y around 0 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-/r* 68.6%

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

   7. Simplified 68.6%

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

Alternative 12: 91.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.1%

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

   1. distribute-rgt-out-- 92.9%

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

3. Simplified 92.9%

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

   1. distribute-rgt-out-- 90.1%

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

   2. associate-/l* 89.6%

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

   3. *-commutative 89.6%

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

   4. distribute-rgt-out-- 92.4%

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

6. Applied egg-rr 92.4%

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

7. Final simplification 92.4%

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

Alternative 13: 91.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.1%

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

   1. distribute-rgt-out-- 92.9%

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

3. Simplified 92.9%

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

   1. add-sqr-sqrt 45.9%

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

   2. *-commutative 45.9%

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

   3. times-frac 45.3%

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

6. Applied egg-rr 45.3%

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

7. Taylor expanded in x around 0 92.1%

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

   1. unpow2 92.1%

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

   2. rem-square-sqrt 92.9%

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

   3. *-commutative 92.9%

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

   4. times-frac 92.2%

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

   5. *-commutative 92.2%

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

   6. associate-*r/ 92.1%

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

   7. *-commutative 92.1%

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

   8. associate-/l* 92.2%

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

9. Simplified 92.2%

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

Alternative 14: 53.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.1%

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

   1. distribute-rgt-out-- 92.9%

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

3. Simplified 92.9%

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

5. Taylor expanded in y around 0 60.8%

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

   1. *-commutative 60.8%

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

7. Simplified 60.8%

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

Developer Target 1: 96.8% 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 2024165 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

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

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