Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 96.4%

Time: 7.9s

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: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m} \cdot x}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4.8e+61)
    (/ x (* (* 0.5 (- y t)) z_m))
    (/ (* (/ 2.0 z_m) x) (- y t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.8e+61) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 4.8d+61) then
        tmp = x / ((0.5d0 * (y - t)) * z_m)
    else
        tmp = ((2.0d0 / z_m) * x) / (y - t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.8e+61) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 4.8e+61:
		tmp = x / ((0.5 * (y - t)) * z_m)
	else:
		tmp = ((2.0 / z_m) * x) / (y - t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.8e+61)
		tmp = Float64(x / Float64(Float64(0.5 * Float64(y - t)) * z_m));
	else
		tmp = Float64(Float64(Float64(2.0 / z_m) * x) / Float64(y - t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 4.8e+61)
		tmp = x / ((0.5 * (y - t)) * z_m);
	else
		tmp = ((2.0 / z_m) * x) / (y - t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 4.8e+61], N[(x / N[(N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / z$95$m), $MachinePrecision] * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.7999999999999998e61

   1. Initial program 90.5%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 93.4

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

   4. Applied rewrites 93.4%

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

   if 4.7999999999999998e61 < z

   1. Initial program 73.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 95.9

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

   4. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot x}{y - t}}{z}} \]
   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. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.5e+72)
    (/ x (* (* 0.5 (- y t)) z_m))
    (* (/ 2.0 (- y t)) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+72) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (2.0 / (y - t)) * (x / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 3.5d+72) then
        tmp = x / ((0.5d0 * (y - t)) * z_m)
    else
        tmp = (2.0d0 / (y - t)) * (x / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+72) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (2.0 / (y - t)) * (x / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 3.5e+72:
		tmp = x / ((0.5 * (y - t)) * z_m)
	else:
		tmp = (2.0 / (y - t)) * (x / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.5e+72)
		tmp = Float64(x / Float64(Float64(0.5 * Float64(y - t)) * z_m));
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 3.5e+72)
		tmp = x / ((0.5 * (y - t)) * z_m);
	else
		tmp = (2.0 / (y - t)) * (x / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 3.5e+72], N[(x / N[(N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000001e72

   1. Initial program 90.5%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 93.4

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

   4. Applied rewrites 93.4%

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

   if 3.5000000000000001e72 < z

   1. Initial program 73.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 97.7

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

   4. Applied rewrites 97.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y \cdot z\_m} \cdot 2\\ \mathbf{elif}\;y \leq 260:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot y\right) \cdot z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -4.5e+130)
    (* (/ x (* y z_m)) 2.0)
    (if (<= y 260.0) (* (/ x (* t z_m)) -2.0) (/ x (* (* 0.5 y) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -4.5e+130) {
		tmp = (x / (y * z_m)) * 2.0;
	} else if (y <= 260.0) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = x / ((0.5 * y) * z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.5d+130)) then
        tmp = (x / (y * z_m)) * 2.0d0
    else if (y <= 260.0d0) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else
        tmp = x / ((0.5d0 * y) * z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -4.5e+130) {
		tmp = (x / (y * z_m)) * 2.0;
	} else if (y <= 260.0) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = x / ((0.5 * y) * z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -4.5e+130:
		tmp = (x / (y * z_m)) * 2.0
	elif y <= 260.0:
		tmp = (x / (t * z_m)) * -2.0
	else:
		tmp = x / ((0.5 * y) * z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -4.5e+130)
		tmp = Float64(Float64(x / Float64(y * z_m)) * 2.0);
	elseif (y <= 260.0)
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	else
		tmp = Float64(x / Float64(Float64(0.5 * y) * z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -4.5e+130)
		tmp = (x / (y * z_m)) * 2.0;
	elseif (y <= 260.0)
		tmp = (x / (t * z_m)) * -2.0;
	else
		tmp = x / ((0.5 * y) * z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -4.5e+130], N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 260.0], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(x / N[(N[(0.5 * y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.50000000000000039e130

   1. Initial program 87.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   if -4.50000000000000039e130 < y < 260

   1. Initial program 89.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   if 260 < y

   1. Initial program 82.5%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 80.2

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

   7. Applied rewrites 80.2%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0.5 \cdot y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{elif}\;y \leq 260:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{y \cdot z\_m} \cdot 2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 260:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x (* y z_m)) 2.0)))
   (*
    z_s
    (if (<= y -4.5e+130) t_1 (if (<= y 260.0) (* (/ x (* t z_m)) -2.0) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (y * z_m)) * 2.0;
	double tmp;
	if (y <= -4.5e+130) {
		tmp = t_1;
	} else if (y <= 260.0) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / (y * z_m)) * 2.0d0
    if (y <= (-4.5d+130)) then
        tmp = t_1
    else if (y <= 260.0d0) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (y * z_m)) * 2.0;
	double tmp;
	if (y <= -4.5e+130) {
		tmp = t_1;
	} else if (y <= 260.0) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / (y * z_m)) * 2.0
	tmp = 0
	if y <= -4.5e+130:
		tmp = t_1
	elif y <= 260.0:
		tmp = (x / (t * z_m)) * -2.0
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(y * z_m)) * 2.0)
	tmp = 0.0
	if (y <= -4.5e+130)
		tmp = t_1;
	elseif (y <= 260.0)
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / (y * z_m)) * 2.0;
	tmp = 0.0;
	if (y <= -4.5e+130)
		tmp = t_1;
	elseif (y <= 260.0)
		tmp = (x / (t * z_m)) * -2.0;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(z$95$s * If[LessEqual[y, -4.5e+130], t$95$1, If[LessEqual[y, 260.0], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000039e130 or 260 < y

   1. Initial program 84.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 79.6

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

   5. Applied rewrites 79.6%

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

   if -4.50000000000000039e130 < y < 260

   1. Initial program 89.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{elif}\;y \leq 260:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{x}{t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t 2.6e+196) (/ x (* (* 0.5 (- y t)) z_m)) (/ (* -2.0 (/ x t)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 2.6e+196) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (-2.0 * (x / t)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.6d+196) then
        tmp = x / ((0.5d0 * (y - t)) * z_m)
    else
        tmp = ((-2.0d0) * (x / t)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 2.6e+196) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (-2.0 * (x / t)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= 2.6e+196:
		tmp = x / ((0.5 * (y - t)) * z_m)
	else:
		tmp = (-2.0 * (x / t)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= 2.6e+196)
		tmp = Float64(x / Float64(Float64(0.5 * Float64(y - t)) * z_m));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / t)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= 2.6e+196)
		tmp = x / ((0.5 * (y - t)) * z_m);
	else
		tmp = (-2.0 * (x / t)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, 2.6e+196], N[(x / N[(N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.60000000000000012e196

   1. Initial program 88.4%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 91.1

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

   4. Applied rewrites 91.1%

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

   if 2.60000000000000012e196 < t

   1. Initial program 77.8%

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

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z\_m} \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t 2.6e+196) (/ x (* (* 0.5 (- y t)) z_m)) (* (/ -2.0 z_m) (/ x t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 2.6e+196) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (-2.0 / z_m) * (x / t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.6d+196) then
        tmp = x / ((0.5d0 * (y - t)) * z_m)
    else
        tmp = ((-2.0d0) / z_m) * (x / t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 2.6e+196) {
		tmp = x / ((0.5 * (y - t)) * z_m);
	} else {
		tmp = (-2.0 / z_m) * (x / t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= 2.6e+196:
		tmp = x / ((0.5 * (y - t)) * z_m)
	else:
		tmp = (-2.0 / z_m) * (x / t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= 2.6e+196)
		tmp = Float64(x / Float64(Float64(0.5 * Float64(y - t)) * z_m));
	else
		tmp = Float64(Float64(-2.0 / z_m) * Float64(x / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= 2.6e+196)
		tmp = x / ((0.5 * (y - t)) * z_m);
	else
		tmp = (-2.0 / z_m) * (x / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, 2.6e+196], N[(x / N[(N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / z$95$m), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.60000000000000012e196

   1. Initial program 88.4%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 91.1

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

   4. Applied rewrites 91.1%

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

   if 2.60000000000000012e196 < t

   1. Initial program 77.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z\_m} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (* z_s (/ x (* (* 0.5 (- y t)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (x / ((0.5 * (y - t)) * z_m));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x / ((0.5d0 * (y - t)) * z_m))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	return z_s * (x / ((0.5 * (y - t)) * z_m));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	return z_s * (x / ((0.5 * (y - t)) * z_m))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	return Float64(z_s * Float64(x / Float64(Float64(0.5 * Float64(y - t)) * z_m)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * (x / ((0.5 * (y - t)) * z_m));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(x / N[(N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 90.1

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

4. Applied rewrites 90.1%

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

5. Final simplification 90.1%

   \[\leadsto \frac{x}{\left(0.5 \cdot \left(y - t\right)\right) \cdot z} \]
6. Add Preprocessing

Alternative 8: 91.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 90.7

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

4. Applied rewrites 90.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift--.f64 N/A

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

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

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

   6. lower-/.f64 N/A

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

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

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 89.7

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

6. Applied rewrites 89.7%

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

7. Final simplification 89.7%

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

Alternative 9: 52.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 52.9

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

5. Applied rewrites 52.9%

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

Alternative 10: 52.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 52.9

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

5. Applied rewrites 52.9%

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

7. Applied rewrites 52.9%

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

8. Final simplification 52.9%

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

Developer Target 1: 96.9% 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 2024294 
(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))))