Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.7% → 96.9%

Time: 11.3s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\left(y - t\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

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) 5e-30)
    (/ (/ x_m z) (* (- y t) 0.5))
    (/ 2.0 (* z (/ (- y t) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-30) {
		tmp = (x_m / z) / ((y - t) * 0.5);
	} else {
		tmp = 2.0 / (z * ((y - t) / x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-30) then
        tmp = (x_m / z) / ((y - t) * 0.5d0)
    else
        tmp = 2.0d0 / (z * ((y - t) / x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-30) {
		tmp = (x_m / z) / ((y - t) * 0.5);
	} else {
		tmp = 2.0 / (z * ((y - t) / x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-30:
		tmp = (x_m / z) / ((y - t) * 0.5)
	else:
		tmp = 2.0 / (z * ((y - t) / x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-30)
		tmp = Float64(Float64(x_m / z) / Float64(Float64(y - t) * 0.5));
	else
		tmp = Float64(2.0 / Float64(z * Float64(Float64(y - t) / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-30)
		tmp = (x_m / z) / ((y - t) * 0.5);
	else
		tmp = 2.0 / (z * ((y - t) / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-30], N[(N[(x$95$m / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(z * N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 4.99999999999999972e-30

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

      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-- 92.8%

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

      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. Step-by-step derivation

      1. associate-*r/ 86.6%

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

      2. associate-*l/ 90.8%

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

      3. clear-num 90.8%

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

      4. un-div-inv 90.8%

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

      5. div-inv 90.8%

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

      6. metadata-eval 90.8%

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

   6. Applied egg-rr 90.8%

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

   if 4.99999999999999972e-30 < (*.f64 x 2)

   1. Initial program 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*l/ 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-rgt-out-- 86.9%

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

      5. associate-/l/ 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/l/ 86.9%

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

      2. associate-*r/ 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-/l* 86.8%

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

      5. *-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. associate-/l* 84.2%

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

      2. associate-/r/ 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{x}{z}}{\left(y - t\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -0.245 \lor \neg \left(y \leq 3.6\right):\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z}\\ \end{array} \end{array} \]

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 (or (<= y -0.245) (not (<= y 3.6)))
    (* x_m (/ (/ 2.0 y) z))
    (* -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 ((y <= -0.245) || !(y <= 3.6)) {
		tmp = x_m * ((2.0 / y) / z);
	} 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 ((y <= (-0.245d0)) .or. (.not. (y <= 3.6d0))) then
        tmp = x_m * ((2.0d0 / y) / z)
    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 ((y <= -0.245) || !(y <= 3.6)) {
		tmp = x_m * ((2.0 / y) / z);
	} 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 (y <= -0.245) or not (y <= 3.6):
		tmp = x_m * ((2.0 / y) / z)
	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 ((y <= -0.245) || !(y <= 3.6))
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	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 ((y <= -0.245) || ~((y <= 3.6)))
		tmp = x_m * ((2.0 / y) / z);
	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[Or[LessEqual[y, -0.245], N[Not[LessEqual[y, 3.6]], $MachinePrecision]], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $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 y < -0.245 or 3.60000000000000009 < y

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      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-- 92.7%

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

      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 inf 78.8%

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

      1. associate-/r* 79.5%

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

   7. Simplified 79.5%

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

   if -0.245 < y < 3.60000000000000009

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. distribute-rgt-out-- 89.3%

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

      5. associate-/l/ 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. associate-/r* 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.245 \lor \neg \left(y \leq 3.6\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -0.028:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 0.003:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \end{array} \end{array} \]

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 (<= y -0.028)
    (* x_m (/ (/ 2.0 y) z))
    (if (<= y 0.003) (* -2.0 (/ (/ x_m t) z)) (/ (* x_m 2.0) (* z y))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -0.028) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 0.003) {
		tmp = -2.0 * ((x_m / t) / z);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-0.028d0)) then
        tmp = x_m * ((2.0d0 / y) / z)
    else if (y <= 0.003d0) then
        tmp = (-2.0d0) * ((x_m / t) / z)
    else
        tmp = (x_m * 2.0d0) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -0.028) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 0.003) {
		tmp = -2.0 * ((x_m / t) / z);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -0.028:
		tmp = x_m * ((2.0 / y) / z)
	elif y <= 0.003:
		tmp = -2.0 * ((x_m / t) / z)
	else:
		tmp = (x_m * 2.0) / (z * y)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -0.028)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	elseif (y <= 0.003)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -0.028)
		tmp = x_m * ((2.0 / y) / z);
	elseif (y <= 0.003)
		tmp = -2.0 * ((x_m / t) / z);
	else
		tmp = (x_m * 2.0) / (z * y);
	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[y, -0.028], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.003], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -0.0280000000000000006

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.2%

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

      3. *-commutative 88.2%

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

      4. distribute-rgt-out-- 89.6%

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

      5. associate-/l/ 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around inf 79.1%

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

      1. associate-/r* 80.3%

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

   7. Simplified 80.3%

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

   if -0.0280000000000000006 < y < 0.0030000000000000001

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. distribute-rgt-out-- 89.3%

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

      5. associate-/l/ 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. associate-/r* 81.1%

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

   7. Simplified 81.1%

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

   if 0.0030000000000000001 < y

   1. Initial program 94.8%

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

      1. distribute-rgt-out-- 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in y around inf 78.6%

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

      1. *-commutative 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.028:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 0.003:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\frac{x\_m}{y \cdot 0.5}}{z}\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \end{array} \end{array} \]

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 (<= y -0.095)
    (/ (/ x_m (* y 0.5)) z)
    (if (<= y 0.0014) (* -2.0 (/ (/ x_m t) z)) (/ (* x_m 2.0) (* z y))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -0.095) {
		tmp = (x_m / (y * 0.5)) / z;
	} else if (y <= 0.0014) {
		tmp = -2.0 * ((x_m / t) / z);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-0.095d0)) then
        tmp = (x_m / (y * 0.5d0)) / z
    else if (y <= 0.0014d0) then
        tmp = (-2.0d0) * ((x_m / t) / z)
    else
        tmp = (x_m * 2.0d0) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -0.095) {
		tmp = (x_m / (y * 0.5)) / z;
	} else if (y <= 0.0014) {
		tmp = -2.0 * ((x_m / t) / z);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -0.095:
		tmp = (x_m / (y * 0.5)) / z
	elif y <= 0.0014:
		tmp = -2.0 * ((x_m / t) / z)
	else:
		tmp = (x_m * 2.0) / (z * y)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(Float64(x_m / Float64(y * 0.5)) / z);
	elseif (y <= 0.0014)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -0.095)
		tmp = (x_m / (y * 0.5)) / z;
	elseif (y <= 0.0014)
		tmp = -2.0 * ((x_m / t) / z);
	else
		tmp = (x_m * 2.0) / (z * y);
	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[y, -0.095], N[(N[(x$95$m / N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 0.0014], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.2%

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

      3. *-commutative 88.2%

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

      4. distribute-rgt-out-- 89.6%

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

      5. associate-/l/ 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around inf 79.1%

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

      1. associate-/r* 80.3%

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

   7. Simplified 80.3%

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

      1. associate-*r/ 83.1%

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

      2. clear-num 83.1%

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

      3. un-div-inv 83.2%

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

      4. div-inv 83.2%

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

      5. metadata-eval 83.2%

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

   9. Applied egg-rr 83.2%

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

   if -0.095000000000000001 < y < 0.00139999999999999999

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. distribute-rgt-out-- 89.3%

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

      5. associate-/l/ 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. associate-/r* 81.1%

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

   7. Simplified 81.1%

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

   if 0.00139999999999999999 < y

   1. Initial program 94.8%

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

      1. distribute-rgt-out-- 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in y around inf 78.6%

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

      1. *-commutative 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\frac{x}{y \cdot 0.5}}{z}\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{+37}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 1e+37)
    (* (/ x_m z) (/ 2.0 (- 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) <= 1e+37) {
		tmp = (x_m / z) * (2.0 / (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) <= 1d+37) then
        tmp = (x_m / z) * (2.0d0 / (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) <= 1e+37) {
		tmp = (x_m / z) * (2.0 / (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) <= 1e+37:
		tmp = (x_m / z) * (2.0 / (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) <= 1e+37)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / 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) <= 1e+37)
		tmp = (x_m / z) * (2.0 / (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], 1e+37], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / 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) < 9.99999999999999954e36

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 93.6%

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

      2. times-frac 91.2%

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

   3. Simplified 91.2%

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

   if 9.99999999999999954e36 < (*.f64 x 2)

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 85.2%

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

   3. Simplified 85.2%

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

      1. *-commutative 85.2%

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

      2. times-frac 98.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{+37}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

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) 5e-30)
    (* (/ x_m z) (/ 2.0 (- y t)))
    (/ 2.0 (* z (/ (- y t) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-30) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = 2.0 / (z * ((y - t) / x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-30) then
        tmp = (x_m / z) * (2.0d0 / (y - t))
    else
        tmp = 2.0d0 / (z * ((y - t) / x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-30) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = 2.0 / (z * ((y - t) / x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-30:
		tmp = (x_m / z) * (2.0 / (y - t))
	else:
		tmp = 2.0 / (z * ((y - t) / x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-30)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(2.0 / Float64(z * Float64(Float64(y - t) / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-30)
		tmp = (x_m / z) * (2.0 / (y - t));
	else
		tmp = 2.0 / (z * ((y - t) / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-30], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(z * N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 4.99999999999999972e-30

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 93.3%

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

      2. times-frac 90.8%

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

   3. Simplified 90.8%

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

   if 4.99999999999999972e-30 < (*.f64 x 2)

   1. Initial program 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*l/ 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-rgt-out-- 86.9%

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

      5. associate-/l/ 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/l/ 86.9%

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

      2. associate-*r/ 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-/l* 86.8%

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

      5. *-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. associate-/l* 84.2%

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

      2. associate-/r/ 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.46 \cdot 10^{-49}:\\ \;\;\;\;x\_m \cdot \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_1\\ \end{array} \end{array} \end{array} \]

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 1.46e-49) (* x_m (/ t_1 z)) (* (/ x_m z) t_1)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 1.46e-49) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = (x_m / z) * t_1;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z <= 1.46d-49) then
        tmp = x_m * (t_1 / z)
    else
        tmp = (x_m / z) * t_1
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 1.46e-49) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = (x_m / z) * t_1;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z <= 1.46e-49:
		tmp = x_m * (t_1 / z)
	else:
		tmp = (x_m / z) * t_1
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z <= 1.46e-49)
		tmp = Float64(x_m * Float64(t_1 / z));
	else
		tmp = Float64(Float64(x_m / z) * t_1);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z <= 1.46e-49)
		tmp = x_m * (t_1 / z);
	else
		tmp = (x_m / z) * t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, 1.46e-49], N[(x$95$m * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.46000000000000007e-49

   1. Initial program 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l/ 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-rgt-out-- 93.2%

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

      5. associate-/l/ 93.5%

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

   3. Simplified 93.5%

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

   if 1.46000000000000007e-49 < z

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 85.1%

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

      2. times-frac 98.2%

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

   3. Simplified 98.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.46 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-51}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z}\\ \end{array} \end{array} \]

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 (<= x_m 4e-51) (* x_m (/ -2.0 (* 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 (x_m <= 4e-51) {
		tmp = x_m * (-2.0 / (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 (x_m <= 4d-51) then
        tmp = x_m * ((-2.0d0) / (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 (x_m <= 4e-51) {
		tmp = x_m * (-2.0 / (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 x_m <= 4e-51:
		tmp = x_m * (-2.0 / (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 (x_m <= 4e-51)
		tmp = Float64(x_m * Float64(-2.0 / 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 (x_m <= 4e-51)
		tmp = x_m * (-2.0 / (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[x$95$m, 4e-51], N[(x$95$m * N[(-2.0 / 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 x < 4e-51

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*l/ 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-rgt-out-- 92.7%

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

      5. associate-/l/ 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around 0 49.3%

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

      1. *-commutative 49.3%

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

   7. Simplified 49.3%

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

   if 4e-51 < x

   1. Initial program 87.4%

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

      1. *-commutative 87.4%

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

      2. associate-*l/ 87.3%

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

      3. *-commutative 87.3%

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

      4. distribute-rgt-out-- 87.5%

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

      5. associate-/l/ 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around 0 55.1%

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

      1. associate-/r* 61.7%

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

   7. Simplified 61.7%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 1.2× speedup

Math

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

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 90.2%

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

   1. *-commutative 90.2%

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

   2. associate-*l/ 89.9%

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

   3. *-commutative 89.9%

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

   4. distribute-rgt-out-- 91.1%

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

   5. associate-/l/ 92.3%

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

3. Simplified 92.3%

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

5. Final simplification 92.3%

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

Alternative 10: 52.9% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* -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) {
	return x_s * (-2.0 * ((x_m / 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 * ((-2.0d0) * ((x_m / 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 * (-2.0 * ((x_m / 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 * (-2.0 * ((x_m / 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(-2.0 * Float64(Float64(x_m / 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 * (-2.0 * ((x_m / 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[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

   1. *-commutative 90.2%

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

   2. associate-*l/ 89.9%

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

   3. *-commutative 89.9%

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

   4. distribute-rgt-out-- 91.1%

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

   5. associate-/l/ 92.3%

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

3. Simplified 92.3%

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

5. Taylor expanded in y around 0 51.3%

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

   1. associate-/r* 52.2%

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

7. Simplified 52.2%

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

8. Final simplification 52.2%

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

Developer target: 97.3% 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 2024040 
(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))))