Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 96.8%

Time: 7.8s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot x\_m \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x\_m}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

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 (<= (* 2.0 x_m) 4e-58)
    (/ (* (/ 2.0 z) x_m) (- 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 ((2.0 * x_m) <= 4e-58) {
		tmp = ((2.0 / z) * x_m) / (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 ((2.0d0 * x_m) <= 4d-58) then
        tmp = ((2.0d0 / z) * x_m) / (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 ((2.0 * x_m) <= 4e-58) {
		tmp = ((2.0 / z) * x_m) / (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 (2.0 * x_m) <= 4e-58:
		tmp = ((2.0 / z) * x_m) / (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(2.0 * x_m) <= 4e-58)
		tmp = Float64(Float64(Float64(2.0 / z) * x_m) / 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 ((2.0 * x_m) <= 4e-58)
		tmp = ((2.0 / z) * x_m) / (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[(2.0 * x$95$m), $MachinePrecision], 4e-58], N[(N[(N[(2.0 / z), $MachinePrecision] * x$95$m), $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)) < 4.0000000000000001e-58

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 94.7

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

   4. Applied rewrites 94.7%

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

   if 4.0000000000000001e-58 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot x\_m \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x\_m}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y - t}{x\_m} \cdot z}\\ \end{array} \end{array} \]

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 (<= (* 2.0 x_m) 4e-58)
    (/ (* (/ 2.0 z) x_m) (- y t))
    (/ 2.0 (* (/ (- y t) x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((2.0 * x_m) <= 4e-58) {
		tmp = ((2.0 / z) * x_m) / (y - t);
	} else {
		tmp = 2.0 / (((y - t) / x_m) * z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((2.0d0 * x_m) <= 4d-58) then
        tmp = ((2.0d0 / z) * x_m) / (y - t)
    else
        tmp = 2.0d0 / (((y - t) / x_m) * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((2.0 * x_m) <= 4e-58) {
		tmp = ((2.0 / z) * x_m) / (y - t);
	} else {
		tmp = 2.0 / (((y - t) / x_m) * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (2.0 * x_m) <= 4e-58:
		tmp = ((2.0 / z) * x_m) / (y - t)
	else:
		tmp = 2.0 / (((y - t) / x_m) * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(2.0 * x_m) <= 4e-58)
		tmp = Float64(Float64(Float64(2.0 / z) * x_m) / Float64(y - t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(y - t) / x_m) * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((2.0 * x_m) <= 4e-58)
		tmp = ((2.0 / z) * x_m) / (y - t);
	else
		tmp = 2.0 / (((y - t) / x_m) * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(2.0 * x$95$m), $MachinePrecision], 4e-58], N[(N[(N[(2.0 / z), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000001e-58

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 94.7

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

   4. Applied rewrites 94.7%

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

   if 4.0000000000000001e-58 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 97.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.3

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

   6. Applied rewrites 97.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y - t}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot x\_m \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y - t}{x\_m} \cdot z}\\ \end{array} \end{array} \]

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 (<= (* 2.0 x_m) 4e-58)
    (* (/ 2.0 (- y t)) (/ x_m z))
    (/ 2.0 (* (/ (- y t) x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((2.0 * x_m) <= 4e-58) {
		tmp = (2.0 / (y - t)) * (x_m / z);
	} else {
		tmp = 2.0 / (((y - t) / x_m) * z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((2.0d0 * x_m) <= 4d-58) then
        tmp = (2.0d0 / (y - t)) * (x_m / z)
    else
        tmp = 2.0d0 / (((y - t) / x_m) * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((2.0 * x_m) <= 4e-58) {
		tmp = (2.0 / (y - t)) * (x_m / z);
	} else {
		tmp = 2.0 / (((y - t) / x_m) * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (2.0 * x_m) <= 4e-58:
		tmp = (2.0 / (y - t)) * (x_m / z)
	else:
		tmp = 2.0 / (((y - t) / x_m) * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(2.0 * x_m) <= 4e-58)
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x_m / z));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(y - t) / x_m) * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((2.0 * x_m) <= 4e-58)
		tmp = (2.0 / (y - t)) * (x_m / z);
	else
		tmp = 2.0 / (((y - t) / x_m) * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(2.0 * x$95$m), $MachinePrecision], 4e-58], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000001e-58

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 94.6

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

   4. Applied rewrites 94.6%

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

   if 4.0000000000000001e-58 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 97.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.3

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

   6. Applied rewrites 97.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y - t}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+24}:\\ \;\;\;\;\frac{t\_1}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (* x_s (if (<= z 8e+24) (* (/ t_1 z) x_m) (* t_1 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 8e+24) {
		tmp = (t_1 / z) * x_m;
	} else {
		tmp = t_1 * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z <= 8d+24) then
        tmp = (t_1 / z) * x_m
    else
        tmp = t_1 * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 8e+24) {
		tmp = (t_1 / z) * x_m;
	} else {
		tmp = t_1 * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z <= 8e+24:
		tmp = (t_1 / z) * x_m
	else:
		tmp = t_1 * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z <= 8e+24)
		tmp = Float64(Float64(t_1 / z) * x_m);
	else
		tmp = Float64(t_1 * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z <= 8e+24)
		tmp = (t_1 / z) * x_m;
	else
		tmp = t_1 * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, 8e+24], N[(N[(t$95$1 / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(t$95$1 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.9999999999999999e24

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

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

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-/.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if 7.9999999999999999e24 < z

   1. Initial program 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 96.5

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

   4. Applied rewrites 96.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{-9}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= z 1e-9)
    (* (/ -2.0 (* (- t y) z)) x_m)
    (* (/ 2.0 (- y t)) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1e-9) {
		tmp = (-2.0 / ((t - y) * z)) * x_m;
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1d-9) then
        tmp = ((-2.0d0) / ((t - y) * z)) * x_m
    else
        tmp = (2.0d0 / (y - t)) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1e-9) {
		tmp = (-2.0 / ((t - y) * z)) * x_m;
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 1e-9:
		tmp = (-2.0 / ((t - y) * z)) * x_m
	else:
		tmp = (2.0 / (y - t)) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 1e-9)
		tmp = Float64(Float64(-2.0 / Float64(Float64(t - y) * z)) * x_m);
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1e-9)
		tmp = (-2.0 / ((t - y) * z)) * x_m;
	else
		tmp = (2.0 / (y - t)) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 1e-9], N[(N[(-2.0 / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000006e-9

   1. Initial program 90.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 92.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 90.8%

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

   if 1.00000000000000006e-9 < z

   1. Initial program 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 96.9

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

   4. Applied rewrites 96.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-9}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2 \cdot x\_m}{y \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (* 2.0 x_m) (* y z))))
   (*
    x_s
    (if (<= y -1.8e-89)
      t_1
      (if (<= y 4.7e-57) (* (/ x_m (* t z)) -2.0) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (2.0 * x_m) / (y * z);
	double tmp;
	if (y <= -1.8e-89) {
		tmp = t_1;
	} else if (y <= 4.7e-57) {
		tmp = (x_m / (t * z)) * -2.0;
	} 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 = (2.0d0 * x_m) / (y * z)
    if (y <= (-1.8d-89)) then
        tmp = t_1
    else if (y <= 4.7d-57) then
        tmp = (x_m / (t * z)) * (-2.0d0)
    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 = (2.0 * x_m) / (y * z);
	double tmp;
	if (y <= -1.8e-89) {
		tmp = t_1;
	} else if (y <= 4.7e-57) {
		tmp = (x_m / (t * z)) * -2.0;
	} 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 = (2.0 * x_m) / (y * z)
	tmp = 0
	if y <= -1.8e-89:
		tmp = t_1
	elif y <= 4.7e-57:
		tmp = (x_m / (t * z)) * -2.0
	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(2.0 * x_m) / Float64(y * z))
	tmp = 0.0
	if (y <= -1.8e-89)
		tmp = t_1;
	elseif (y <= 4.7e-57)
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	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 = (2.0 * x_m) / (y * z);
	tmp = 0.0;
	if (y <= -1.8e-89)
		tmp = t_1;
	elseif (y <= 4.7e-57)
		tmp = (x_m / (t * z)) * -2.0;
	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[(2.0 * x$95$m), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.8e-89], t$95$1, If[LessEqual[y, 4.7e-57], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000003e-89 or 4.6999999999999998e-57 < y

   1. Initial program 90.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -1.80000000000000003e-89 < y < 4.6999999999999998e-57

   1. Initial program 88.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{y \cdot z} \cdot 2\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* y z)) 2.0)))
   (*
    x_s
    (if (<= y -1.8e-89)
      t_1
      (if (<= y 4.7e-57) (* (/ x_m (* t z)) -2.0) 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 / (y * z)) * 2.0;
	double tmp;
	if (y <= -1.8e-89) {
		tmp = t_1;
	} else if (y <= 4.7e-57) {
		tmp = (x_m / (t * z)) * -2.0;
	} 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 / (y * z)) * 2.0d0
    if (y <= (-1.8d-89)) then
        tmp = t_1
    else if (y <= 4.7d-57) then
        tmp = (x_m / (t * z)) * (-2.0d0)
    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 / (y * z)) * 2.0;
	double tmp;
	if (y <= -1.8e-89) {
		tmp = t_1;
	} else if (y <= 4.7e-57) {
		tmp = (x_m / (t * z)) * -2.0;
	} 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 / (y * z)) * 2.0
	tmp = 0
	if y <= -1.8e-89:
		tmp = t_1
	elif y <= 4.7e-57:
		tmp = (x_m / (t * z)) * -2.0
	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 / Float64(y * z)) * 2.0)
	tmp = 0.0
	if (y <= -1.8e-89)
		tmp = t_1;
	elseif (y <= 4.7e-57)
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	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 / (y * z)) * 2.0;
	tmp = 0.0;
	if (y <= -1.8e-89)
		tmp = t_1;
	elseif (y <= 4.7e-57)
		tmp = (x_m / (t * z)) * -2.0;
	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 / N[(y * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.8e-89], t$95$1, If[LessEqual[y, 4.7e-57], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000003e-89 or 4.6999999999999998e-57 < y

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.80000000000000003e-89 < y < 4.6999999999999998e-57

   1. Initial program 88.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.2× speedup

Math

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

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 (* (- t y) z)) 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) {
	return x_s * ((-2.0 / ((t - y) * z)) * x_m);
}

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) / ((t - y) * z)) * x_m)
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 / ((t - y) * z)) * x_m);
}

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 / ((t - y) * z)) * x_m)

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

Derivation

1. Initial program 89.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 93.5

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

4. Applied rewrites 93.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 93.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 93.1

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

6. Applied rewrites 93.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

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

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

8. Applied rewrites 91.6%

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

Alternative 9: 53.7% accurate, 1.4× speedup

Math

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

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 (* t z)) -2.0)))

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 / (t * z)) * -2.0);
}

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 / (t * z)) * (-2.0d0))
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 / (t * z)) * -2.0);
}

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 / (t * z)) * -2.0)

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

Derivation

1. Initial program 89.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 49.9

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

5. Applied rewrites 49.9%

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

Alternative 10: 53.5% accurate, 1.4× speedup

Math

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

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 (* t z)) 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) {
	return x_s * ((-2.0 / (t * z)) * x_m);
}

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) / (t * z)) * x_m)
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 / (t * z)) * x_m);
}

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 / (t * z)) * x_m)

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

Derivation

1. Initial program 89.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 49.9

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

5. Applied rewrites 49.9%

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

7. Applied rewrites 49.8%

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

8. Final simplification 49.8%

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

Developer Target 1: 96.7% 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 2024295 
(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))))