Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.9% → 96.1%

Time: 10.0s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% 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: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{x \cdot 2}{z_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4.2e+110)
    (/ (* x 2.0) (* z_m (- y t)))
    (* (/ x z_m) (/ 2.0 (- y t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.2e+110) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 4.2d+110) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.2e+110) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 4.2e+110:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (x / z_m) * (2.0 / (y - t))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.2e+110)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 4.2e+110)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (x / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 4.2e+110], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.2000000000000003e110

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   if 4.2000000000000003e110 < z

   1. Initial program 75.1%

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

      1. distribute-rgt-out-- 83.9%

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

      2. times-frac 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{-6} \lor \neg \left(t \leq 4 \cdot 10^{-112}\right):\\ \;\;\;\;x \cdot \frac{-2}{z_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -3.65e-6) (not (<= t 4e-112)))
    (* x (/ -2.0 (* z_m t)))
    (* x (/ 2.0 (* z_m y))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -3.65e-6) || !(t <= 4e-112)) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = x * (2.0 / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-3.65d-6)) .or. (.not. (t <= 4d-112))) then
        tmp = x * ((-2.0d0) / (z_m * t))
    else
        tmp = x * (2.0d0 / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -3.65e-6) || !(t <= 4e-112)) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = x * (2.0 / (z_m * y));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (t <= -3.65e-6) or not (t <= 4e-112):
		tmp = x * (-2.0 / (z_m * t))
	else:
		tmp = x * (2.0 / (z_m * y))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((t <= -3.65e-6) || !(t <= 4e-112))
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((t <= -3.65e-6) || ~((t <= 4e-112)))
		tmp = x * (-2.0 / (z_m * t));
	else
		tmp = x * (2.0 / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[t, -3.65e-6], N[Not[LessEqual[t, 4e-112]], $MachinePrecision]], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.65000000000000021e-6 or 3.9999999999999998e-112 < t

   1. Initial program 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-*l/ 91.2%

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

      3. *-commutative 91.2%

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

      4. distribute-rgt-out-- 93.1%

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

      5. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 73.4%

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

      1. *-commutative 73.4%

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

   7. Simplified 73.4%

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

   if -3.65000000000000021e-6 < t < 3.9999999999999998e-112

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l/ 92.6%

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

      3. *-commutative 92.6%

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

      4. distribute-rgt-out-- 94.5%

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

      5. associate-/l/ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around inf 82.3%

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

      1. *-commutative 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{-6} \lor \neg \left(t \leq 4 \cdot 10^{-112}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -0.00056 \lor \neg \left(t \leq 3.5 \cdot 10^{-112}\right):\\ \;\;\;\;x \cdot \frac{-2}{z_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -0.00056) (not (<= t 3.5e-112)))
    (* x (/ -2.0 (* z_m t)))
    (* x (/ (/ 2.0 y) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -0.00056) || !(t <= 3.5e-112)) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = x * ((2.0 / y) / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.00056d0)) .or. (.not. (t <= 3.5d-112))) then
        tmp = x * ((-2.0d0) / (z_m * t))
    else
        tmp = x * ((2.0d0 / y) / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -0.00056) || !(t <= 3.5e-112)) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = x * ((2.0 / y) / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (t <= -0.00056) or not (t <= 3.5e-112):
		tmp = x * (-2.0 / (z_m * t))
	else:
		tmp = x * ((2.0 / y) / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((t <= -0.00056) || !(t <= 3.5e-112))
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(x * Float64(Float64(2.0 / y) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((t <= -0.00056) || ~((t <= 3.5e-112)))
		tmp = x * (-2.0 / (z_m * t));
	else
		tmp = x * ((2.0 / y) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[t, -0.00056], N[Not[LessEqual[t, 3.5e-112]], $MachinePrecision]], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -5.5999999999999995e-4 or 3.49999999999999994e-112 < t

   1. Initial program 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-*l/ 91.2%

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

      3. *-commutative 91.2%

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

      4. distribute-rgt-out-- 93.1%

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

      5. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 73.4%

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

      1. *-commutative 73.4%

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

   7. Simplified 73.4%

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

   if -5.5999999999999995e-4 < t < 3.49999999999999994e-112

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l/ 92.6%

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

      3. *-commutative 92.6%

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

      4. distribute-rgt-out-- 94.5%

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

      5. associate-/l/ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around inf 82.3%

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

      1. associate-/r* 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00056 \lor \neg \left(t \leq 3.5 \cdot 10^{-112}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;-2 \cdot \frac{x}{z_m \cdot t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z_m \cdot t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1e+103)
    (* -2.0 (/ x (* z_m t)))
    (if (<= t 4.8e-20) (* (/ x z_m) (/ 2.0 y)) (* x (/ -2.0 (* z_m t)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1e+103) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 4.8e-20) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = x * (-2.0 / (z_m * t));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d+103)) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else if (t <= 4.8d-20) then
        tmp = (x / z_m) * (2.0d0 / y)
    else
        tmp = x * ((-2.0d0) / (z_m * t))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1e+103) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 4.8e-20) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = x * (-2.0 / (z_m * t));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -1e+103:
		tmp = -2.0 * (x / (z_m * t))
	elif t <= 4.8e-20:
		tmp = (x / z_m) * (2.0 / y)
	else:
		tmp = x * (-2.0 / (z_m * t))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -1e+103)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	elseif (t <= 4.8e-20)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -1e+103)
		tmp = -2.0 * (x / (z_m * t));
	elseif (t <= 4.8e-20)
		tmp = (x / z_m) * (2.0 / y);
	else
		tmp = x * (-2.0 / (z_m * t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -1e+103], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e-20], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1e103

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      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/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -1e103 < t < 4.79999999999999986e-20

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 93.4%

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

      2. times-frac 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in y around inf 78.4%

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

   if 4.79999999999999986e-20 < t

   1. Initial program 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*l/ 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-rgt-out-- 94.7%

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

      5. associate-/l/ 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around 0 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+103}:\\ \;\;\;\;-2 \cdot \frac{x}{z_m \cdot t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z_m \cdot t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1.2e+103)
    (* -2.0 (/ x (* z_m t)))
    (if (<= t 1.45e-20) (/ (/ (* x 2.0) z_m) y) (* x (/ -2.0 (* z_m t)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.2e+103) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 1.45e-20) {
		tmp = ((x * 2.0) / z_m) / y;
	} else {
		tmp = x * (-2.0 / (z_m * t));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.2d+103)) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else if (t <= 1.45d-20) then
        tmp = ((x * 2.0d0) / z_m) / y
    else
        tmp = x * ((-2.0d0) / (z_m * t))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.2e+103) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 1.45e-20) {
		tmp = ((x * 2.0) / z_m) / y;
	} else {
		tmp = x * (-2.0 / (z_m * t));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -1.2e+103:
		tmp = -2.0 * (x / (z_m * t))
	elif t <= 1.45e-20:
		tmp = ((x * 2.0) / z_m) / y
	else:
		tmp = x * (-2.0 / (z_m * t))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -1.2e+103)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	elseif (t <= 1.45e-20)
		tmp = Float64(Float64(Float64(x * 2.0) / z_m) / y);
	else
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.2e+103)
		tmp = -2.0 * (x / (z_m * t));
	elseif (t <= 1.45e-20)
		tmp = ((x * 2.0) / z_m) / y;
	else
		tmp = x * (-2.0 / (z_m * t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -1.2e+103], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e-20], N[(N[(N[(x * 2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.1999999999999999e103

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      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/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -1.1999999999999999e103 < t < 1.45e-20

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*l/ 91.9%

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

      3. *-commutative 91.9%

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

      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

   5. Taylor expanded in y around inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. *-commutative 73.8%

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

      3. *-commutative 73.8%

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

      4. associate-/r* 78.5%

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

      5. *-commutative 78.5%

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

   7. Simplified 78.5%

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

   if 1.45e-20 < t

   1. Initial program 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*l/ 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-rgt-out-- 94.7%

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

      5. associate-/l/ 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around 0 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+103}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z_m}}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y 1.6e+168) (* x (/ (/ 2.0 (- y t)) z_m)) (/ (/ (* x 2.0) z_m) y))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 1.6e+168) {
		tmp = x * ((2.0 / (y - t)) / z_m);
	} else {
		tmp = ((x * 2.0) / z_m) / y;
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.6d+168) then
        tmp = x * ((2.0d0 / (y - t)) / z_m)
    else
        tmp = ((x * 2.0d0) / z_m) / y
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 1.6e+168) {
		tmp = x * ((2.0 / (y - t)) / z_m);
	} else {
		tmp = ((x * 2.0) / z_m) / y;
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= 1.6e+168:
		tmp = x * ((2.0 / (y - t)) / z_m)
	else:
		tmp = ((x * 2.0) / z_m) / y
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= 1.6e+168)
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y - t)) / z_m));
	else
		tmp = Float64(Float64(Float64(x * 2.0) / z_m) / y);
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= 1.6e+168)
		tmp = x * ((2.0 / (y - t)) / z_m);
	else
		tmp = ((x * 2.0) / z_m) / y;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, 1.6e+168], N[(x * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.6000000000000001e168

   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/ 94.2%

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

      3. *-commutative 94.2%

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

      4. distribute-rgt-out-- 96.0%

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

      5. associate-/l/ 96.1%

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

   3. Simplified 96.1%

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

   if 1.6000000000000001e168 < y

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l/ 73.5%

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

      3. *-commutative 73.5%

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

      4. distribute-rgt-out-- 76.8%

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

      5. associate-/l/ 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in y around inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. *-commutative 73.8%

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

      3. *-commutative 73.8%

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

      4. associate-/r* 90.6%

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

      5. *-commutative 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{t_1}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot t_1\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (* z_s (if (<= z_m 2e+41) (* x (/ t_1 z_m)) (* (/ x z_m) t_1)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 2e+41) {
		tmp = x * (t_1 / z_m);
	} else {
		tmp = (x / z_m) * t_1;
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z_m <= 2d+41) then
        tmp = x * (t_1 / z_m)
    else
        tmp = (x / z_m) * t_1
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 2e+41) {
		tmp = x * (t_1 / z_m);
	} else {
		tmp = (x / z_m) * t_1;
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z_m <= 2e+41:
		tmp = x * (t_1 / z_m)
	else:
		tmp = (x / z_m) * t_1
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z_m <= 2e+41)
		tmp = Float64(x * Float64(t_1 / z_m));
	else
		tmp = Float64(Float64(x / z_m) * t_1);
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z_m <= 2e+41)
		tmp = x * (t_1 / z_m);
	else
		tmp = (x / z_m) * t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 2e+41], N[(x * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000001e41

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

      3. *-commutative 94.4%

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

      4. distribute-rgt-out-- 95.3%

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

      5. associate-/l/ 95.3%

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

   3. Simplified 95.3%

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

   if 2.00000000000000001e41 < z

   1. Initial program 79.9%

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

      1. distribute-rgt-out-- 86.3%

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

      2. times-frac 98.8%

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

   3. Simplified 98.8%

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

4. Final simplification 96.0%

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

Alternative 8: 53.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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/ 91.7%

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

   3. *-commutative 91.7%

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

   4. distribute-rgt-out-- 93.7%

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

   5. associate-/l/ 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 52.4%

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

   1. *-commutative 52.4%

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

7. Simplified 52.4%

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

8. Final simplification 52.4%

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

Alternative 9: 53.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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/ 91.7%

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

   3. *-commutative 91.7%

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

   4. distribute-rgt-out-- 93.7%

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

   5. associate-/l/ 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 52.5%

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

   1. *-commutative 52.5%

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

7. Simplified 52.5%

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

8. Final simplification 52.5%

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

Developer target: 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 2024020 
(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))))