Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.0% → 96.9%

Time: 9.9s

Alternatives: 9

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: 90.0% 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: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \cdot 2 \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\frac{x_m \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2e+69)
    (/ (* x_m 2.0) (* z (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e+69) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 2d+69) then
        tmp = (x_m * 2.0d0) / (z * (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e+69) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 2e+69:
		tmp = (x_m * 2.0) / (z * (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 2e+69)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 2e+69)
		tmp = (x_m * 2.0) / (z * (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 2e+69], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 2.0000000000000001e69

   1. Initial program 93.7%

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

      1. distribute-rgt-out-- 96.2%

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

   3. Simplified 96.2%

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

   if 2.0000000000000001e69 < (*.f64 x 2)

   1. Initial program 81.1%

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

      1. distribute-rgt-out-- 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      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 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;x_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -4.2e-32)
    (* -2.0 (/ (/ x_m t) z))
    (if (<= t 1.3e-53) (* x_m (/ (/ 2.0 y) z)) (* 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) {
	double tmp;
	if (t <= -4.2e-32) {
		tmp = -2.0 * ((x_m / t) / z);
	} else if (t <= 1.3e-53) {
		tmp = x_m * ((2.0 / y) / z);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 (t <= (-4.2d-32)) then
        tmp = (-2.0d0) * ((x_m / t) / z)
    else if (t <= 1.3d-53) then
        tmp = x_m * ((2.0d0 / y) / z)
    else
        tmp = x_m * (((-2.0d0) / t) / 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 (t <= -4.2e-32) {
		tmp = -2.0 * ((x_m / t) / z);
	} else if (t <= 1.3e-53) {
		tmp = x_m * ((2.0 / y) / z);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 t <= -4.2e-32:
		tmp = -2.0 * ((x_m / t) / z)
	elif t <= 1.3e-53:
		tmp = x_m * ((2.0 / y) / z)
	else:
		tmp = x_m * ((-2.0 / t) / 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 (t <= -4.2e-32)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z));
	elseif (t <= 1.3e-53)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / 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 (t <= -4.2e-32)
		tmp = -2.0 * ((x_m / t) / z);
	elseif (t <= 1.3e-53)
		tmp = x_m * ((2.0 / y) / z);
	else
		tmp = x_m * ((-2.0 / t) / 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[t, -4.2e-32], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-53], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.1999999999999998e-32

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. distribute-rgt-out-- 91.8%

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

      5. associate-/l/ 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. associate-/r* 77.9%

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

   7. Simplified 77.9%

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

   if -4.1999999999999998e-32 < t < 1.29999999999999998e-53

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l/ 93.9%

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

      3. *-commutative 93.9%

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

      4. distribute-rgt-out-- 95.0%

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

      5. associate-/l/ 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around inf 84.5%

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

      1. associate-/r* 84.4%

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

   7. Simplified 84.4%

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

   if 1.29999999999999998e-53 < t

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-*l/ 91.1%

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

      3. *-commutative 91.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. associate-/l/ 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 77.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-60}:\\ \;\;\;\;\frac{x_m \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -5.4e-32)
    (* -2.0 (/ (/ x_m t) z))
    (if (<= t 2.45e-60) (/ (* x_m 2.0) (* z y)) (* 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) {
	double tmp;
	if (t <= -5.4e-32) {
		tmp = -2.0 * ((x_m / t) / z);
	} else if (t <= 2.45e-60) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 (t <= (-5.4d-32)) then
        tmp = (-2.0d0) * ((x_m / t) / z)
    else if (t <= 2.45d-60) then
        tmp = (x_m * 2.0d0) / (z * y)
    else
        tmp = x_m * (((-2.0d0) / t) / 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 (t <= -5.4e-32) {
		tmp = -2.0 * ((x_m / t) / z);
	} else if (t <= 2.45e-60) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 t <= -5.4e-32:
		tmp = -2.0 * ((x_m / t) / z)
	elif t <= 2.45e-60:
		tmp = (x_m * 2.0) / (z * y)
	else:
		tmp = x_m * ((-2.0 / t) / 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 (t <= -5.4e-32)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z));
	elseif (t <= 2.45e-60)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / 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 (t <= -5.4e-32)
		tmp = -2.0 * ((x_m / t) / z);
	elseif (t <= 2.45e-60)
		tmp = (x_m * 2.0) / (z * y);
	else
		tmp = x_m * ((-2.0 / t) / 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[t, -5.4e-32], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.45e-60], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.39999999999999962e-32

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. distribute-rgt-out-- 91.8%

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

      5. associate-/l/ 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. associate-/r* 77.9%

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

   7. Simplified 77.9%

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

   if -5.39999999999999962e-32 < t < 2.44999999999999994e-60

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in y around inf 84.6%

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

      1. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   if 2.44999999999999994e-60 < t

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-*l/ 91.1%

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

      3. *-commutative 91.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. associate-/l/ 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 77.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-60}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{x_m}{t} \cdot -2}{z}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x_m \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -2.6e-32)
    (/ (* (/ x_m t) -2.0) z)
    (if (<= t 2.2e-54) (/ (* x_m 2.0) (* z y)) (* 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) {
	double tmp;
	if (t <= -2.6e-32) {
		tmp = ((x_m / t) * -2.0) / z;
	} else if (t <= 2.2e-54) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 (t <= (-2.6d-32)) then
        tmp = ((x_m / t) * (-2.0d0)) / z
    else if (t <= 2.2d-54) then
        tmp = (x_m * 2.0d0) / (z * y)
    else
        tmp = x_m * (((-2.0d0) / t) / 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 (t <= -2.6e-32) {
		tmp = ((x_m / t) * -2.0) / z;
	} else if (t <= 2.2e-54) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * ((-2.0 / t) / 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 t <= -2.6e-32:
		tmp = ((x_m / t) * -2.0) / z
	elif t <= 2.2e-54:
		tmp = (x_m * 2.0) / (z * y)
	else:
		tmp = x_m * ((-2.0 / t) / 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 (t <= -2.6e-32)
		tmp = Float64(Float64(Float64(x_m / t) * -2.0) / z);
	elseif (t <= 2.2e-54)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / 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 (t <= -2.6e-32)
		tmp = ((x_m / t) * -2.0) / z;
	elseif (t <= 2.2e-54)
		tmp = (x_m * 2.0) / (z * y);
	else
		tmp = x_m * ((-2.0 / t) / 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[t, -2.6e-32], N[(N[(N[(x$95$m / t), $MachinePrecision] * -2.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 2.2e-54], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -2.5999999999999997e-32

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. distribute-rgt-out-- 91.8%

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

      5. associate-/l/ 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-/r* 77.9%

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

      3. associate-*l/ 78.0%

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

   7. Applied egg-rr 78.0%

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

   if -2.5999999999999997e-32 < t < 2.2e-54

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in y around inf 84.6%

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

      1. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   if 2.2e-54 < t

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-*l/ 91.1%

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

      3. *-commutative 91.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. associate-/l/ 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 77.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{x}{t} \cdot -2}{z}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \cdot 2 \leq 2 \cdot 10^{+69}:\\ \;\;\;\;x_m \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2e+69)
    (* 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) <= 2e+69) {
		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) <= 2d+69) 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) <= 2e+69) {
		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) <= 2e+69:
		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) <= 2e+69)
		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) <= 2e+69)
		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], 2e+69], 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 2) < 2.0000000000000001e69

   1. Initial program 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l/ 93.5%

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

      3. *-commutative 93.5%

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

      4. distribute-rgt-out-- 96.0%

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

      5. associate-/l/ 96.4%

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

   3. Simplified 96.4%

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

   if 2.0000000000000001e69 < (*.f64 x 2)

   1. Initial program 81.1%

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

      1. distribute-rgt-out-- 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      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 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;x_m \cdot \frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{x_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (* x_s (if (<= z 6.2e-47) (* x_m (/ t_1 z)) (* t_1 (/ x_m 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 (z <= 6.2e-47) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = t_1 * (x_m / 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 (z <= 6.2d-47) then
        tmp = x_m * (t_1 / z)
    else
        tmp = t_1 * (x_m / 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 (z <= 6.2e-47) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = t_1 * (x_m / 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 z <= 6.2e-47:
		tmp = x_m * (t_1 / z)
	else:
		tmp = t_1 * (x_m / 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 (z <= 6.2e-47)
		tmp = Float64(x_m * Float64(t_1 / z));
	else
		tmp = Float64(t_1 * Float64(x_m / 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 (z <= 6.2e-47)
		tmp = x_m * (t_1 / z);
	else
		tmp = t_1 * (x_m / 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[z, 6.2e-47], N[(x$95$m * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.1999999999999996e-47

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-*l/ 90.9%

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

      3. *-commutative 90.9%

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

      4. distribute-rgt-out-- 93.3%

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

      5. associate-/l/ 93.6%

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

   3. Simplified 93.6%

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

   if 6.1999999999999996e-47 < z

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 93.0%

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

      2. times-frac 97.9%

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

   3. Simplified 97.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5000000000:\\ \;\;\;\;-2 \cdot \frac{x_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 5000000000.0) (* -2.0 (/ x_m (* z t))) (* -2.0 (/ (/ x_m 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) {
	double tmp;
	if (z <= 5000000000.0) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / t) / 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 <= 5000000000.0d0) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else
        tmp = (-2.0d0) * ((x_m / t) / 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 <= 5000000000.0) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / t) / 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 <= 5000000000.0:
		tmp = -2.0 * (x_m / (z * t))
	else:
		tmp = -2.0 * ((x_m / t) / 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 <= 5000000000.0)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / 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 <= 5000000000.0)
		tmp = -2.0 * (x_m / (z * t));
	else
		tmp = -2.0 * ((x_m / t) / 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, 5000000000.0], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5e9

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l/ 91.5%

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

      3. *-commutative 91.5%

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

      4. distribute-rgt-out-- 93.7%

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

      5. associate-/l/ 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around 0 58.0%

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

   if 5e9 < z

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*l/ 88.6%

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

      3. *-commutative 88.6%

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

      4. distribute-rgt-out-- 91.5%

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

      5. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in y around 0 47.1%

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

      1. associate-/r* 51.6%

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

   7. Simplified 51.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5000000000:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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(Float64(2.0 / 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[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

   1. *-commutative 91.0%

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

   2. associate-*l/ 90.7%

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

   3. *-commutative 90.7%

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

   4. distribute-rgt-out-- 93.1%

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

   5. associate-/l/ 93.8%

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

3. Simplified 93.8%

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

5. Final simplification 93.8%

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

Alternative 9: 53.4% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* -2.0 (/ x_m (* z t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. *-commutative 91.0%

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

   2. associate-*l/ 90.7%

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

   3. *-commutative 90.7%

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

   4. distribute-rgt-out-- 93.1%

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

   5. associate-/l/ 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in y around 0 55.1%

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

6. Final simplification 55.1%

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

Developer target: 97.2% 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 2024019 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

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

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