Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 97.0%

Time: 10.1s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% 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, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m \cdot \frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e-15)
    (/ (* x_m (/ 2.0 z)) (- 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) <= 4e-15) {
		tmp = (x_m * (2.0 / z)) / (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) <= 4d-15) then
        tmp = (x_m * (2.0d0 / z)) / (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) <= 4e-15) {
		tmp = (x_m * (2.0 / z)) / (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) <= 4e-15:
		tmp = (x_m * (2.0 / z)) / (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) <= 4e-15)
		tmp = Float64(Float64(x_m * Float64(2.0 / z)) / Float64(y - t));
	else
		tmp = Float64(Float64(2.0 / 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) <= 4e-15)
		tmp = (x_m * (2.0 / z)) / (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], 4e-15], N[(N[(x$95$m * N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000003e-15

   1. Initial program 89.7%

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

      1. *-commutative N/A

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

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

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 92.2

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

   4. Applied egg-rr 92.2%

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

   if 4.0000000000000003e-15 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 83.5%

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

      1. clear-num N/A

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

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

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. --lowering--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.7× speedup

Math

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

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-7)
    (/ (* x_m (/ 2.0 z)) (- y t))
    (/ (/ (* 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) {
	double tmp;
	if ((x_m * 2.0) <= 2e-7) {
		tmp = (x_m * (2.0 / z)) / (y - t);
	} else {
		tmp = ((x_m * 2.0) / (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-7) then
        tmp = (x_m * (2.0d0 / z)) / (y - t)
    else
        tmp = ((x_m * 2.0d0) / (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-7) {
		tmp = (x_m * (2.0 / z)) / (y - t);
	} else {
		tmp = ((x_m * 2.0) / (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-7:
		tmp = (x_m * (2.0 / z)) / (y - t)
	else:
		tmp = ((x_m * 2.0) / (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-7)
		tmp = Float64(Float64(x_m * Float64(2.0 / z)) / Float64(y - t));
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / 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-7)
		tmp = (x_m * (2.0 / z)) / (y - t);
	else
		tmp = ((x_m * 2.0) / (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-7], N[(N[(x$95$m * N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.9999999999999999e-7

   1. Initial program 89.7%

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

      1. *-commutative N/A

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

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

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 92.2

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

   4. Applied egg-rr 92.2%

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

   if 1.9999999999999999e-7 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 83.5%

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

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 94.1%

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

Alternative 3: 96.7% accurate, 0.7× speedup

Math

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

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 (<= (* x_m 2.0) 1e-6)
    (/ (* x_m (/ 2.0 z)) (- y t))
    (* (/ 2.0 z) (/ x_m (- 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 ((x_m * 2.0) <= 1e-6) {
		tmp = (x_m * (2.0 / z)) / (y - t);
	} else {
		tmp = (2.0 / z) * (x_m / (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 ((x_m * 2.0d0) <= 1d-6) then
        tmp = (x_m * (2.0d0 / z)) / (y - t)
    else
        tmp = (2.0d0 / z) * (x_m / (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 ((x_m * 2.0) <= 1e-6) {
		tmp = (x_m * (2.0 / z)) / (y - t);
	} else {
		tmp = (2.0 / z) * (x_m / (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 (x_m * 2.0) <= 1e-6:
		tmp = (x_m * (2.0 / z)) / (y - t)
	else:
		tmp = (2.0 / z) * (x_m / (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 (Float64(x_m * 2.0) <= 1e-6)
		tmp = Float64(Float64(x_m * Float64(2.0 / z)) / Float64(y - t));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / 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 ((x_m * 2.0) <= 1e-6)
		tmp = (x_m * (2.0 / z)) / (y - t);
	else
		tmp = (2.0 / z) * (x_m / (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[N[(x$95$m * 2.0), $MachinePrecision], 1e-6], N[(N[(x$95$m * N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 89.8%

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

      1. *-commutative N/A

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

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

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 92.3

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

   4. Applied egg-rr 92.3%

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

   if 9.99999999999999955e-7 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 83.2%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 99.5

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-6}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y - t}\\ \end{array} \end{array} \]

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 1.22e+67)
    (* x_m (/ 2.0 (* z (- y t))))
    (* (/ 2.0 z) (/ x_m (- 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 <= 1.22e+67) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (2.0 / z) * (x_m / (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 <= 1.22d+67) then
        tmp = x_m * (2.0d0 / (z * (y - t)))
    else
        tmp = (2.0d0 / z) * (x_m / (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 <= 1.22e+67) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (2.0 / z) * (x_m / (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 <= 1.22e+67:
		tmp = x_m * (2.0 / (z * (y - t)))
	else:
		tmp = (2.0 / z) * (x_m / (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 <= 1.22e+67)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * Float64(y - t))));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / 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 <= 1.22e+67)
		tmp = x_m * (2.0 / (z * (y - t)));
	else
		tmp = (2.0 / z) * (x_m / (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, 1.22e+67], N[(x$95$m * N[(2.0 / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.22000000000000004e67

   1. Initial program 91.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. --lowering--.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   if 1.22000000000000004e67 < z

   1. Initial program 72.7%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 97.8

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot -2}{z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m -2.0) (* z t))))
   (*
    x_s
    (if (<= t -2.25e-150)
      t_1
      (if (<= t 2.05e+116) (/ (* x_m 2.0) (* z y)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * -2.0) / (z * t);
	double tmp;
	if (t <= -2.25e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (-2.0d0)) / (z * t)
    if (t <= (-2.25d-150)) then
        tmp = t_1
    else if (t <= 2.05d+116) then
        tmp = (x_m * 2.0d0) / (z * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * -2.0) / (z * t);
	double tmp;
	if (t <= -2.25e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * -2.0) / (z * t)
	tmp = 0
	if t <= -2.25e-150:
		tmp = t_1
	elif t <= 2.05e+116:
		tmp = (x_m * 2.0) / (z * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * -2.0) / Float64(z * t))
	tmp = 0.0
	if (t <= -2.25e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * -2.0) / (z * t);
	tmp = 0.0;
	if (t <= -2.25e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = (x_m * 2.0) / (z * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.25e-150], t$95$1, If[LessEqual[t, 2.05e+116], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2500000000000001e-150 or 2.0499999999999999e116 < t

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 77.2

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

   5. Simplified 77.2%

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

   if -2.2500000000000001e-150 < t < 2.0499999999999999e116

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 80.3

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

   5. Simplified 80.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-150}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot -2}{z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m -2.0) (* z t))))
   (*
    x_s
    (if (<= t -2.9e-150)
      t_1
      (if (<= t 2.05e+116) (* x_m (/ 2.0 (* z y))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * -2.0) / (z * t);
	double tmp;
	if (t <= -2.9e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (-2.0d0)) / (z * t)
    if (t <= (-2.9d-150)) then
        tmp = t_1
    else if (t <= 2.05d+116) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * -2.0) / (z * t);
	double tmp;
	if (t <= -2.9e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * -2.0) / (z * t)
	tmp = 0
	if t <= -2.9e-150:
		tmp = t_1
	elif t <= 2.05e+116:
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * -2.0) / Float64(z * t))
	tmp = 0.0
	if (t <= -2.9e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * -2.0) / (z * t);
	tmp = 0.0;
	if (t <= -2.9e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.9e-150], t$95$1, If[LessEqual[t, 2.05e+116], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999998e-150 or 2.0499999999999999e116 < t

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 77.2

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

   5. Simplified 77.2%

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

   if -2.8999999999999998e-150 < t < 2.0499999999999999e116

   1. Initial program 90.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. --lowering--.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 80.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{-2}{z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ -2.0 (* z t)))))
   (*
    x_s
    (if (<= t -6.6e-150)
      t_1
      (if (<= t 2.05e+116) (* x_m (/ 2.0 (* z y))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (-2.0 / (z * t));
	double tmp;
	if (t <= -6.6e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((-2.0d0) / (z * t))
    if (t <= (-6.6d-150)) then
        tmp = t_1
    else if (t <= 2.05d+116) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (-2.0 / (z * t));
	double tmp;
	if (t <= -6.6e-150) {
		tmp = t_1;
	} else if (t <= 2.05e+116) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (-2.0 / (z * t))
	tmp = 0
	if t <= -6.6e-150:
		tmp = t_1
	elif t <= 2.05e+116:
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(-2.0 / Float64(z * t)))
	tmp = 0.0
	if (t <= -6.6e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (-2.0 / (z * t));
	tmp = 0.0;
	if (t <= -6.6e-150)
		tmp = t_1;
	elseif (t <= 2.05e+116)
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -6.6e-150], t$95$1, If[LessEqual[t, 2.05e+116], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6000000000000003e-150 or 2.0499999999999999e116 < t

   1. Initial program 86.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. --lowering--.f64 92.4

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

   4. Applied egg-rr 92.4%

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

   5. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.4

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

   7. Simplified 77.4%

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

   if -6.6000000000000003e-150 < t < 2.0499999999999999e116

   1. Initial program 89.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. --lowering--.f64 92.3

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 79.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 88.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. --lowering--.f64 92.4

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

4. Applied egg-rr 92.4%

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

5. Final simplification 92.4%

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

Alternative 9: 53.1% accurate, 1.4× speedup

Math

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

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 (* x_m (/ -2.0 (* 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 * (x_m * (-2.0 / (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 * (x_m * ((-2.0d0) / (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 * (x_m * (-2.0 / (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 * (x_m * (-2.0 / (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(x_m * Float64(-2.0 / 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 * (x_m * (-2.0 / (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[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. --lowering--.f64 92.4

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

4. Applied egg-rr 92.4%

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

5. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 52.5

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

7. Simplified 52.5%

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

8. Final simplification 52.5%

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

Developer Target 1: 96.8% 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 2024204 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (* x 2) (- (* y z) (* t z))) -2559141628295061/10000000000000000000000000000) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 522513913665063/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2))))

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