Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 88.8% → 95.7%

Time: 10.0s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% 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: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x\_m \cdot \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_1}{z}\\ \end{array} \end{array} \end{array} \]

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 (- y t))))
   (* x_s (if (<= (* x_m 2.0) 4e+104) (* x_m (/ t_1 z)) (/ (* x_m t_1) 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 t_1 = 2.0 / (y - t);
	double tmp;
	if ((x_m * 2.0) <= 4e+104) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = (x_m * t_1) / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if ((x_m * 2.0d0) <= 4d+104) then
        tmp = x_m * (t_1 / z)
    else
        tmp = (x_m * t_1) / 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 t_1 = 2.0 / (y - t);
	double tmp;
	if ((x_m * 2.0) <= 4e+104) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = (x_m * t_1) / 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):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if (x_m * 2.0) <= 4e+104:
		tmp = x_m * (t_1 / z)
	else:
		tmp = (x_m * t_1) / 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)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+104)
		tmp = Float64(x_m * Float64(t_1 / z));
	else
		tmp = Float64(Float64(x_m * t_1) / 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)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+104)
		tmp = x_m * (t_1 / z);
	else
		tmp = (x_m * t_1) / 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_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+104], N[(x$95$m * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$1), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4e104

   1. Initial program 95.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 95.6

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

   4. Applied egg-rr 95.6%

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. --lowering--.f64 96.3

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

   6. Applied egg-rr 96.3%

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

   if 4e104 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- N/A

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

      2. times-frac N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 97.6

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

   4. Applied egg-rr 97.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y - t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

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) 4e+104)
    (* x_m (/ (/ 2.0 (- y t)) z))
    (* (/ 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) <= 4e+104) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} 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) <= 4d+104) then
        tmp = x_m * ((2.0d0 / (y - t)) / z)
    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) <= 4e+104) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} 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) <= 4e+104:
		tmp = x_m * ((2.0 / (y - t)) / z)
	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) <= 4e+104)
		tmp = Float64(x_m * Float64(Float64(2.0 / Float64(y - t)) / z));
	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) <= 4e+104)
		tmp = x_m * ((2.0 / (y - t)) / z);
	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], 4e+104], N[(x$95$m * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $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)) < 4e104

   1. Initial program 95.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 95.6

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

   4. Applied egg-rr 95.6%

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. --lowering--.f64 96.3

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

   6. Applied egg-rr 96.3%

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

   if 4e104 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 97.5

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{x\_m \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

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 (<= z 1.8e-27)
    (/ (* x_m 2.0) (* (- y t) z))
    (* (/ 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 (z <= 1.8e-27) {
		tmp = (x_m * 2.0) / ((y - t) * z);
	} 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 (z <= 1.8d-27) then
        tmp = (x_m * 2.0d0) / ((y - t) * z)
    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 (z <= 1.8e-27) {
		tmp = (x_m * 2.0) / ((y - t) * z);
	} 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 z <= 1.8e-27:
		tmp = (x_m * 2.0) / ((y - t) * z)
	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 (z <= 1.8e-27)
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z));
	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 (z <= 1.8e-27)
		tmp = (x_m * 2.0) / ((y - t) * z);
	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[z, 1.8e-27], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $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 z < 1.7999999999999999e-27

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

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

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

      4. --lowering--.f64 94.1

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

   4. Applied egg-rr 94.1%

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

   if 1.7999999999999999e-27 < z

   1. Initial program 94.4%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 98.4

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

   4. Applied egg-rr 98.4%

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

Alternative 4: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(z \cdot -0.5\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \end{array} \end{array} \]

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 (<= t -1.65e-5)
    (/ x_m (* t (* z -0.5)))
    (if (<= t 5.8e-21) (/ (* x_m 2.0) (* y z)) (* (/ x_m (* t z)) -2.0)))))

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 (t <= -1.65e-5) {
		tmp = x_m / (t * (z * -0.5));
	} else if (t <= 5.8e-21) {
		tmp = (x_m * 2.0) / (y * z);
	} else {
		tmp = (x_m / (t * z)) * -2.0;
	}
	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 (t <= (-1.65d-5)) then
        tmp = x_m / (t * (z * (-0.5d0)))
    else if (t <= 5.8d-21) then
        tmp = (x_m * 2.0d0) / (y * z)
    else
        tmp = (x_m / (t * z)) * (-2.0d0)
    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 (t <= -1.65e-5) {
		tmp = x_m / (t * (z * -0.5));
	} else if (t <= 5.8e-21) {
		tmp = (x_m * 2.0) / (y * z);
	} else {
		tmp = (x_m / (t * z)) * -2.0;
	}
	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 t <= -1.65e-5:
		tmp = x_m / (t * (z * -0.5))
	elif t <= 5.8e-21:
		tmp = (x_m * 2.0) / (y * z)
	else:
		tmp = (x_m / (t * z)) * -2.0
	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 (t <= -1.65e-5)
		tmp = Float64(x_m / Float64(t * Float64(z * -0.5)));
	elseif (t <= 5.8e-21)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	else
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	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 (t <= -1.65e-5)
		tmp = x_m / (t * (z * -0.5));
	elseif (t <= 5.8e-21)
		tmp = (x_m * 2.0) / (y * z);
	else
		tmp = (x_m / (t * z)) * -2.0;
	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[t, -1.65e-5], N[(x$95$m / N[(t * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e-21], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.6500000000000001e-5

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 83.1

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

   5. Simplified 83.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval 83.1

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

   7. Applied egg-rr 83.1%

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

   if -1.6500000000000001e-5 < t < 5.8e-21

   1. Initial program 96.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 82.0

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

   5. Simplified 82.0%

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

   if 5.8e-21 < t

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 75.2

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

   5. Simplified 75.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 75.2

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

   7. Applied egg-rr 75.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{t \cdot \left(z \cdot -0.5\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(z \cdot -0.5\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \end{array} \end{array} \]

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 (<= t -1.65e-6)
    (/ x_m (* t (* z -0.5)))
    (if (<= t 1.9e-20) (* x_m (/ 2.0 (* y z))) (* (/ x_m (* t z)) -2.0)))))

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 (t <= -1.65e-6) {
		tmp = x_m / (t * (z * -0.5));
	} else if (t <= 1.9e-20) {
		tmp = x_m * (2.0 / (y * z));
	} else {
		tmp = (x_m / (t * z)) * -2.0;
	}
	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 (t <= (-1.65d-6)) then
        tmp = x_m / (t * (z * (-0.5d0)))
    else if (t <= 1.9d-20) then
        tmp = x_m * (2.0d0 / (y * z))
    else
        tmp = (x_m / (t * z)) * (-2.0d0)
    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 (t <= -1.65e-6) {
		tmp = x_m / (t * (z * -0.5));
	} else if (t <= 1.9e-20) {
		tmp = x_m * (2.0 / (y * z));
	} else {
		tmp = (x_m / (t * z)) * -2.0;
	}
	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 t <= -1.65e-6:
		tmp = x_m / (t * (z * -0.5))
	elif t <= 1.9e-20:
		tmp = x_m * (2.0 / (y * z))
	else:
		tmp = (x_m / (t * z)) * -2.0
	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 (t <= -1.65e-6)
		tmp = Float64(x_m / Float64(t * Float64(z * -0.5)));
	elseif (t <= 1.9e-20)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z)));
	else
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	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 (t <= -1.65e-6)
		tmp = x_m / (t * (z * -0.5));
	elseif (t <= 1.9e-20)
		tmp = x_m * (2.0 / (y * z));
	else
		tmp = (x_m / (t * z)) * -2.0;
	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[t, -1.65e-6], N[(x$95$m / N[(t * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-20], N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.65000000000000008e-6

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 83.1

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

   5. Simplified 83.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval 83.1

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

   7. Applied egg-rr 83.1%

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

   if -1.65000000000000008e-6 < t < 1.8999999999999999e-20

   1. Initial program 96.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 81.9%

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

   if 1.8999999999999999e-20 < t

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 75.2

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

   5. Simplified 75.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 75.2

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

   7. Applied egg-rr 75.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{t \cdot \left(z \cdot -0.5\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t \cdot z} \cdot -2\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* t z)) -2.0)))
   (*
    x_s
    (if (<= t -3.7e-7) t_1 (if (<= t 1.7e-20) (* x_m (/ 2.0 (* y z))) 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 / (t * z)) * -2.0;
	double tmp;
	if (t <= -3.7e-7) {
		tmp = t_1;
	} else if (t <= 1.7e-20) {
		tmp = x_m * (2.0 / (y * z));
	} 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 / (t * z)) * (-2.0d0)
    if (t <= (-3.7d-7)) then
        tmp = t_1
    else if (t <= 1.7d-20) then
        tmp = x_m * (2.0d0 / (y * z))
    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 / (t * z)) * -2.0;
	double tmp;
	if (t <= -3.7e-7) {
		tmp = t_1;
	} else if (t <= 1.7e-20) {
		tmp = x_m * (2.0 / (y * z));
	} 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 / (t * z)) * -2.0
	tmp = 0
	if t <= -3.7e-7:
		tmp = t_1
	elif t <= 1.7e-20:
		tmp = x_m * (2.0 / (y * z))
	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 / Float64(t * z)) * -2.0)
	tmp = 0.0
	if (t <= -3.7e-7)
		tmp = t_1;
	elseif (t <= 1.7e-20)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z)));
	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 / (t * z)) * -2.0;
	tmp = 0.0;
	if (t <= -3.7e-7)
		tmp = t_1;
	elseif (t <= 1.7e-20)
		tmp = x_m * (2.0 / (y * z));
	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 / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -3.7e-7], t$95$1, If[LessEqual[t, 1.7e-20], N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.70000000000000004e-7 or 1.6999999999999999e-20 < t

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 78.7

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

   5. Simplified 78.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

   if -3.70000000000000004e-7 < t < 1.6999999999999999e-20

   1. Initial program 96.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 81.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{-2}{t \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* t z)))))
   (*
    x_s
    (if (<= t -2.9e-8) t_1 (if (<= t 1.25e-20) (* x_m (/ 2.0 (* y z))) 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 / (t * z));
	double tmp;
	if (t <= -2.9e-8) {
		tmp = t_1;
	} else if (t <= 1.25e-20) {
		tmp = x_m * (2.0 / (y * z));
	} 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) / (t * z))
    if (t <= (-2.9d-8)) then
        tmp = t_1
    else if (t <= 1.25d-20) then
        tmp = x_m * (2.0d0 / (y * z))
    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 / (t * z));
	double tmp;
	if (t <= -2.9e-8) {
		tmp = t_1;
	} else if (t <= 1.25e-20) {
		tmp = x_m * (2.0 / (y * z));
	} 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 / (t * z))
	tmp = 0
	if t <= -2.9e-8:
		tmp = t_1
	elif t <= 1.25e-20:
		tmp = x_m * (2.0 / (y * z))
	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(t * z)))
	tmp = 0.0
	if (t <= -2.9e-8)
		tmp = t_1;
	elseif (t <= 1.25e-20)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z)));
	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 / (t * z));
	tmp = 0.0;
	if (t <= -2.9e-8)
		tmp = t_1;
	elseif (t <= 1.25e-20)
		tmp = x_m * (2.0 / (y * z));
	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[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.9e-8], t$95$1, If[LessEqual[t, 1.25e-20], N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9000000000000002e-8 or 1.25e-20 < t

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.6

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

   7. Simplified 78.6%

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

   if -2.9000000000000002e-8 < t < 1.25e-20

   1. Initial program 96.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-out-- N/A

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

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

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

      7. --lowering--.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 81.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 1.2× speedup

Math

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

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 (/ (* x_m 2.0) (* (- y t) 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) {
	return x_s * ((x_m * 2.0) / ((y - t) * z));
}

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 * ((x_m * 2.0d0) / ((y - t) * z))
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 * ((x_m * 2.0) / ((y - t) * z));
}

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

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(Float64(x_m * 2.0) / Float64(Float64(y - t) * z)))
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 * ((x_m * 2.0) / ((y - t) * z));
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[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. distribute-rgt-out-- N/A

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

   2. *-commutative N/A

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

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

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

   4. --lowering--.f64 94.6

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

4. Applied egg-rr 94.6%

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

Alternative 9: 91.0% accurate, 1.2× speedup

Math

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

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 (* x_m (/ 2.0 (* (- y t) 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) {
	return x_s * (x_m * (2.0 / ((y - t) * z)));
}

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 * (x_m * (2.0d0 / ((y - t) * z)))
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 * (x_m * (2.0 / ((y - t) * z)));
}

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

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(x_m * Float64(2.0 / Float64(Float64(y - t) * z))))
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 * (x_m * (2.0 / ((y - t) * z)));
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[(x$95$m * N[(2.0 / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. distribute-rgt-out-- N/A

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

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

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

   7. --lowering--.f64 94.4

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

4. Applied egg-rr 94.4%

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

5. Final simplification 94.4%

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

Alternative 10: 52.3% accurate, 1.4× speedup

Math

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

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 (* x_m (/ -2.0 (* t 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) {
	return x_s * (x_m * (-2.0 / (t * z)));
}

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 * (x_m * ((-2.0d0) / (t * z)))
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 * (x_m * (-2.0 / (t * z)));
}

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 * (x_m * (-2.0 / (t * z)))

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(x_m * Float64(-2.0 / Float64(t * z))))
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 * (x_m * (-2.0 / (t * z)));
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[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. distribute-rgt-out-- N/A

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

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

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

   7. --lowering--.f64 94.4

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

4. Applied egg-rr 94.4%

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

5. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 52.0

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

7. Simplified 52.0%

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

8. Final simplification 52.0%

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

Developer Target 1: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(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))))