Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.1% → 97.0%

Time: 10.3s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 97.0% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2e-49)
    (* 2.0 (/ (/ x_m z) (- y t)))
    (* 2.0 (/ (/ x_m (- 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) {
	double tmp;
	if ((x_m * 2.0) <= 2e-49) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = 2.0 * ((x_m / (y - 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 * 2.0d0) <= 2d-49) then
        tmp = 2.0d0 * ((x_m / z) / (y - t))
    else
        tmp = 2.0d0 * ((x_m / (y - 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 * 2.0) <= 2e-49) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = 2.0 * ((x_m / (y - 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 * 2.0) <= 2e-49:
		tmp = 2.0 * ((x_m / z) / (y - t))
	else:
		tmp = 2.0 * ((x_m / (y - 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 (Float64(x_m * 2.0) <= 2e-49)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / Float64(y - 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 * 2.0) <= 2e-49)
		tmp = 2.0 * ((x_m / z) / (y - t));
	else
		tmp = 2.0 * ((x_m / (y - 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[N[(x$95$m * 2.0), $MachinePrecision], 2e-49], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.99999999999999987e-49

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. associate-/r* 93.7%

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

   7. Simplified 93.7%

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

   if 1.99999999999999987e-49 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. times-frac 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. clear-num 98.0%

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

      2. frac-times 98.3%

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

      3. metadata-eval 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 98.3%

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

      3. associate-/r* 98.3%

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

      4. clear-num 98.4%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 94.9%

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

Alternative 2: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-66} \lor \neg \left(y \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -4.1e-66) (not (<= y 8e-11)))
    (* 2.0 (/ (/ x_m y) z))
    (* -2.0 (/ x_m (* z t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -4.1e-66) || !(y <= 8e-11)) {
		tmp = 2.0 * ((x_m / y) / z);
	} else {
		tmp = -2.0 * (x_m / (z * t));
	}
	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 <= (-4.1d-66)) .or. (.not. (y <= 8d-11))) then
        tmp = 2.0d0 * ((x_m / y) / z)
    else
        tmp = (-2.0d0) * (x_m / (z * t))
    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 <= -4.1e-66) || !(y <= 8e-11)) {
		tmp = 2.0 * ((x_m / y) / z);
	} else {
		tmp = -2.0 * (x_m / (z * t));
	}
	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 <= -4.1e-66) or not (y <= 8e-11):
		tmp = 2.0 * ((x_m / y) / z)
	else:
		tmp = -2.0 * (x_m / (z * t))
	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 <= -4.1e-66) || !(y <= 8e-11))
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / z));
	else
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	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 <= -4.1e-66) || ~((y <= 8e-11)))
		tmp = 2.0 * ((x_m / y) / z);
	else
		tmp = -2.0 * (x_m / (z * t));
	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, -4.1e-66], N[Not[LessEqual[y, 8e-11]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.09999999999999998e-66 or 7.99999999999999952e-11 < y

   1. Initial program 86.8%

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

      1. distribute-rgt-out-- 90.6%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 76.8%

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

      1. associate-/r* 77.1%

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

   9. Simplified 77.1%

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

   if -4.09999999999999998e-66 < y < 7.99999999999999952e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.8%

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

      1. *-commutative 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-66} \lor \neg \left(y \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-67} \lor \neg \left(y \leq 8.2 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -7.8e-67) (not (<= y 8.2e-11)))
    (* 2.0 (/ (/ x_m y) z))
    (* x_m (/ (/ -2.0 t) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -7.8e-67) || !(y <= 8.2e-11)) {
		tmp = 2.0 * ((x_m / y) / z);
	} else {
		tmp = x_m * ((-2.0 / t) / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-7.8d-67)) .or. (.not. (y <= 8.2d-11))) then
        tmp = 2.0d0 * ((x_m / y) / z)
    else
        tmp = x_m * (((-2.0d0) / t) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -7.8e-67) || !(y <= 8.2e-11)) {
		tmp = 2.0 * ((x_m / y) / z);
	} else {
		tmp = x_m * ((-2.0 / t) / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -7.8e-67) or not (y <= 8.2e-11):
		tmp = 2.0 * ((x_m / y) / z)
	else:
		tmp = x_m * ((-2.0 / t) / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -7.8e-67) || !(y <= 8.2e-11))
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / z));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -7.8e-67) || ~((y <= 8.2e-11)))
		tmp = 2.0 * ((x_m / y) / z);
	else
		tmp = x_m * ((-2.0 / t) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -7.8e-67], N[Not[LessEqual[y, 8.2e-11]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -7.7999999999999997e-67 or 8.2000000000000001e-11 < y

   1. Initial program 86.8%

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

      1. distribute-rgt-out-- 90.6%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 76.8%

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

      1. associate-/r* 77.1%

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

   9. Simplified 77.1%

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

   if -7.7999999999999997e-67 < y < 8.2000000000000001e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. clear-num 96.6%

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

      2. frac-times 96.3%

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

      3. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in y around 0 76.8%

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

       1. associate-*r/ 76.8%

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

       2. associate-*l/ 76.8%

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

       3. *-commutative 76.8%

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

       4. associate-/r* 77.3%

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

   11. Simplified 77.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-67} \lor \neg \left(y \leq 8.2 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.7e-65)
    (* x_m (/ 2.0 (* z y)))
    (if (<= y 9.5e-11) (* x_m (/ (/ -2.0 t) z)) (* 2.0 (/ (/ x_m y) 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 <= -3.7e-65) {
		tmp = x_m * (2.0 / (z * y));
	} else if (y <= 9.5e-11) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= (-3.7d-65)) then
        tmp = x_m * (2.0d0 / (z * y))
    else if (y <= 9.5d-11) then
        tmp = x_m * (((-2.0d0) / t) / z)
    else
        tmp = 2.0d0 * ((x_m / y) / 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 <= -3.7e-65) {
		tmp = x_m * (2.0 / (z * y));
	} else if (y <= 9.5e-11) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= -3.7e-65:
		tmp = x_m * (2.0 / (z * y))
	elif y <= 9.5e-11:
		tmp = x_m * ((-2.0 / t) / z)
	else:
		tmp = 2.0 * ((x_m / y) / 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 <= -3.7e-65)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	elseif (y <= 9.5e-11)
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / 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 <= -3.7e-65)
		tmp = x_m * (2.0 / (z * y));
	elseif (y <= 9.5e-11)
		tmp = x_m * ((-2.0 / t) / z);
	else
		tmp = 2.0 * ((x_m / y) / 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[y, -3.7e-65], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-11], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.7e-65

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

      1. distribute-rgt-out-- 85.7%

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

      2. associate-/l* 85.6%

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

      3. *-commutative 85.6%

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

      4. distribute-rgt-out-- 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in y around inf 74.7%

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

      1. *-commutative 74.7%

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

   9. Simplified 74.7%

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

   if -3.7e-65 < y < 9.49999999999999951e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. clear-num 96.6%

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

      2. frac-times 96.3%

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

      3. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in y around 0 76.8%

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

       1. associate-*r/ 76.8%

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

       2. associate-*l/ 76.8%

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

       3. *-commutative 76.8%

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

       4. associate-/r* 77.3%

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

   11. Simplified 77.3%

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

   if 9.49999999999999951e-11 < y

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf 79.0%

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

      1. associate-/r* 80.5%

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

   9. Simplified 80.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.55e-65)
    (* (/ x_m z) (/ 2.0 y))
    (if (<= y 9.2e-11) (* x_m (/ (/ -2.0 t) z)) (* 2.0 (/ (/ x_m y) 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 <= -3.55e-65) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 9.2e-11) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= (-3.55d-65)) then
        tmp = (x_m / z) * (2.0d0 / y)
    else if (y <= 9.2d-11) then
        tmp = x_m * (((-2.0d0) / t) / z)
    else
        tmp = 2.0d0 * ((x_m / y) / 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 <= -3.55e-65) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 9.2e-11) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= -3.55e-65:
		tmp = (x_m / z) * (2.0 / y)
	elif y <= 9.2e-11:
		tmp = x_m * ((-2.0 / t) / z)
	else:
		tmp = 2.0 * ((x_m / y) / 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 <= -3.55e-65)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	elseif (y <= 9.2e-11)
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / 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 <= -3.55e-65)
		tmp = (x_m / z) * (2.0 / y);
	elseif (y <= 9.2e-11)
		tmp = x_m * ((-2.0 / t) / z);
	else
		tmp = 2.0 * ((x_m / y) / 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[y, -3.55e-65], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e-11], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.55000000000000014e-65

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. times-frac 77.3%

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

   9. Applied egg-rr 77.3%

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

   if -3.55000000000000014e-65 < y < 9.20000000000000054e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. clear-num 96.6%

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

      2. frac-times 96.3%

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

      3. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in y around 0 76.8%

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

       1. associate-*r/ 76.8%

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

       2. associate-*l/ 76.8%

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

       3. *-commutative 76.8%

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

       4. associate-/r* 77.3%

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

   11. Simplified 77.3%

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

   if 9.20000000000000054e-11 < y

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf 79.0%

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

      1. associate-/r* 80.5%

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

   9. Simplified 80.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -6.8e-65)
    (* (/ x_m z) (/ 2.0 y))
    (if (<= y 8e-11) (/ -2.0 (* z (/ t x_m))) (* 2.0 (/ (/ x_m y) 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 <= -6.8e-65) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 8e-11) {
		tmp = -2.0 / (z * (t / x_m));
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= (-6.8d-65)) then
        tmp = (x_m / z) * (2.0d0 / y)
    else if (y <= 8d-11) then
        tmp = (-2.0d0) / (z * (t / x_m))
    else
        tmp = 2.0d0 * ((x_m / y) / 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 <= -6.8e-65) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 8e-11) {
		tmp = -2.0 / (z * (t / x_m));
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= -6.8e-65:
		tmp = (x_m / z) * (2.0 / y)
	elif y <= 8e-11:
		tmp = -2.0 / (z * (t / x_m))
	else:
		tmp = 2.0 * ((x_m / y) / 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 <= -6.8e-65)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	elseif (y <= 8e-11)
		tmp = Float64(-2.0 / Float64(z * Float64(t / x_m)));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / 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 <= -6.8e-65)
		tmp = (x_m / z) * (2.0 / y);
	elseif (y <= 8e-11)
		tmp = -2.0 / (z * (t / x_m));
	else
		tmp = 2.0 * ((x_m / y) / 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[y, -6.8e-65], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-11], N[(-2.0 / N[(z * N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6.79999999999999973e-65

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. times-frac 77.3%

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

   9. Applied egg-rr 77.3%

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

   if -6.79999999999999973e-65 < y < 7.99999999999999952e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.8%

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

      1. *-commutative 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

      3. *-commutative 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. *-commutative 76.5%

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

       2. associate-/l* 80.3%

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

   11. Applied egg-rr 80.3%

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

   if 7.99999999999999952e-11 < y

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf 79.0%

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

      1. associate-/r* 80.5%

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

   9. Simplified 80.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{y \cdot 0.5}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -7.3e-65)
    (/ (/ x_m z) (* y 0.5))
    (if (<= y 1.02e-10) (/ -2.0 (* z (/ t x_m))) (* 2.0 (/ (/ x_m y) 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 <= -7.3e-65) {
		tmp = (x_m / z) / (y * 0.5);
	} else if (y <= 1.02e-10) {
		tmp = -2.0 / (z * (t / x_m));
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= (-7.3d-65)) then
        tmp = (x_m / z) / (y * 0.5d0)
    else if (y <= 1.02d-10) then
        tmp = (-2.0d0) / (z * (t / x_m))
    else
        tmp = 2.0d0 * ((x_m / y) / 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 <= -7.3e-65) {
		tmp = (x_m / z) / (y * 0.5);
	} else if (y <= 1.02e-10) {
		tmp = -2.0 / (z * (t / x_m));
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= -7.3e-65:
		tmp = (x_m / z) / (y * 0.5)
	elif y <= 1.02e-10:
		tmp = -2.0 / (z * (t / x_m))
	else:
		tmp = 2.0 * ((x_m / y) / 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 <= -7.3e-65)
		tmp = Float64(Float64(x_m / z) / Float64(y * 0.5));
	elseif (y <= 1.02e-10)
		tmp = Float64(-2.0 / Float64(z * Float64(t / x_m)));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / 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 <= -7.3e-65)
		tmp = (x_m / z) / (y * 0.5);
	elseif (y <= 1.02e-10)
		tmp = -2.0 / (z * (t / x_m));
	else
		tmp = 2.0 * ((x_m / y) / 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[y, -7.3e-65], N[(N[(x$95$m / z), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e-10], N[(-2.0 / N[(z * N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.2999999999999998e-65

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. times-frac 77.3%

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

   9. Applied egg-rr 77.3%

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

       1. clear-num 77.3%

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

       2. un-div-inv 77.4%

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

       3. div-inv 77.4%

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

       4. metadata-eval 77.4%

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

   11. Applied egg-rr 77.4%

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

   if -7.2999999999999998e-65 < y < 1.01999999999999997e-10

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.8%

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

      1. *-commutative 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

      3. *-commutative 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. *-commutative 76.5%

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

       2. associate-/l* 80.3%

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

   11. Applied egg-rr 80.3%

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

   if 1.01999999999999997e-10 < y

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf 79.0%

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

      1. associate-/r* 80.5%

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

   9. Simplified 80.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot 0.5}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{y \cdot 0.5}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x\_m}{t \cdot -0.5}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.8e-65)
    (/ (/ x_m z) (* y 0.5))
    (if (<= y 9.5e-11) (/ (/ x_m (* t -0.5)) z) (* 2.0 (/ (/ x_m y) 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 <= -3.8e-65) {
		tmp = (x_m / z) / (y * 0.5);
	} else if (y <= 9.5e-11) {
		tmp = (x_m / (t * -0.5)) / z;
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= (-3.8d-65)) then
        tmp = (x_m / z) / (y * 0.5d0)
    else if (y <= 9.5d-11) then
        tmp = (x_m / (t * (-0.5d0))) / z
    else
        tmp = 2.0d0 * ((x_m / y) / 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 <= -3.8e-65) {
		tmp = (x_m / z) / (y * 0.5);
	} else if (y <= 9.5e-11) {
		tmp = (x_m / (t * -0.5)) / z;
	} else {
		tmp = 2.0 * ((x_m / y) / 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 <= -3.8e-65:
		tmp = (x_m / z) / (y * 0.5)
	elif y <= 9.5e-11:
		tmp = (x_m / (t * -0.5)) / z
	else:
		tmp = 2.0 * ((x_m / y) / 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 <= -3.8e-65)
		tmp = Float64(Float64(x_m / z) / Float64(y * 0.5));
	elseif (y <= 9.5e-11)
		tmp = Float64(Float64(x_m / Float64(t * -0.5)) / z);
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / y) / 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 <= -3.8e-65)
		tmp = (x_m / z) / (y * 0.5);
	elseif (y <= 9.5e-11)
		tmp = (x_m / (t * -0.5)) / z;
	else
		tmp = 2.0 * ((x_m / y) / 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[y, -3.8e-65], N[(N[(x$95$m / z), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-11], N[(N[(x$95$m / N[(t * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000002e-65

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. times-frac 77.3%

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

   9. Applied egg-rr 77.3%

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

       1. clear-num 77.3%

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

       2. un-div-inv 77.4%

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

       3. div-inv 77.4%

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

       4. metadata-eval 77.4%

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

   11. Applied egg-rr 77.4%

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

   if -3.8000000000000002e-65 < y < 9.49999999999999951e-11

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. clear-num 96.6%

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

      2. frac-times 96.3%

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

      3. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in y around 0 76.8%

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

       1. associate-*r/ 76.8%

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

       2. associate-*l/ 76.8%

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

       3. *-commutative 76.8%

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

       4. associate-/r* 77.3%

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

   11. Simplified 77.3%

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

       1. associate-*r/ 80.5%

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

       2. clear-num 80.5%

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

       3. un-div-inv 80.6%

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

       4. div-inv 80.6%

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

       5. metadata-eval 80.6%

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

   13. Applied egg-rr 80.6%

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

   if 9.49999999999999951e-11 < y

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf 79.0%

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

      1. associate-/r* 80.5%

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

   9. Simplified 80.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot 0.5}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x}{t \cdot -0.5}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2e-49)
    (* 2.0 (/ (/ x_m z) (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e-49) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 2d-49) then
        tmp = 2.0d0 * ((x_m / z) / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2e-49) {
		tmp = 2.0 * ((x_m / z) / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 2e-49:
		tmp = 2.0 * ((x_m / z) / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 2e-49)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 2e-49)
		tmp = 2.0 * ((x_m / z) / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 2e-49], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / 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 #s(literal 2 binary64)) < 1.99999999999999987e-49

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. associate-/r* 93.7%

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

   7. Simplified 93.7%

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

   if 1.99999999999999987e-49 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      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 94.8%

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

Alternative 10: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y - t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 7.5e-62)
    (* x_m (/ 2.0 (* z (- y t))))
    (* 2.0 (/ (/ x_m z) (- y t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 7.5e-62) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = 2.0 * ((x_m / z) / (y - t));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7.5d-62) then
        tmp = x_m * (2.0d0 / (z * (y - t)))
    else
        tmp = 2.0d0 * ((x_m / z) / (y - t))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 7.5e-62) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = 2.0 * ((x_m / z) / (y - t));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 7.5e-62:
		tmp = x_m * (2.0 / (z * (y - t)))
	else:
		tmp = 2.0 * ((x_m / z) / (y - t))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 7.5e-62)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * Float64(y - t))));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / Float64(y - t)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 7.5e-62)
		tmp = x_m * (2.0 / (z * (y - t)));
	else
		tmp = 2.0 * ((x_m / z) / (y - t));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 7.5e-62], N[(x$95$m * N[(2.0 / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5000000000000003e-62

   1. Initial program 89.4%

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

      1. distribute-rgt-out-- 92.2%

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

   3. Simplified 92.2%

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

      1. distribute-rgt-out-- 89.4%

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

      2. associate-/l* 89.3%

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

      3. *-commutative 89.3%

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

      4. distribute-rgt-out-- 92.0%

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

   6. Applied egg-rr 92.0%

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

   if 7.5000000000000003e-62 < z

   1. Initial program 84.0%

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

      1. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in x around 0 86.8%

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

      1. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. distribute-rgt-out-- 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in x around 0 90.6%

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

   1. associate-/r* 91.9%

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

7. Simplified 91.9%

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

8. Final simplification 91.9%

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

Alternative 12: 54.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. distribute-rgt-out-- 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in y around 0 53.3%

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

   1. *-commutative 53.3%

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

7. Simplified 53.3%

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

8. Final simplification 53.3%

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

Developer target: 97.1% 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 2024073 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (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))))