Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.2% → 96.2%

Time: 8.3s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% 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: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{0.5 \cdot \left(y - t\right)}\\ \end{array} \end{array} \]

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 9e+97)
    (* (/ -2.0 (* (- t y) z_m)) x)
    (/ (/ x z_m) (* 0.5 (- 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 <= 9e+97) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} else {
		tmp = (x / z_m) / (0.5 * (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 <= 9d+97) then
        tmp = ((-2.0d0) / ((t - y) * z_m)) * x
    else
        tmp = (x / z_m) / (0.5d0 * (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 <= 9e+97) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} else {
		tmp = (x / z_m) / (0.5 * (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 <= 9e+97:
		tmp = (-2.0 / ((t - y) * z_m)) * x
	else:
		tmp = (x / z_m) / (0.5 * (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 <= 9e+97)
		tmp = Float64(Float64(-2.0 / Float64(Float64(t - y) * z_m)) * x);
	else
		tmp = Float64(Float64(x / z_m) / Float64(0.5 * 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 <= 9e+97)
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	else
		tmp = (x / z_m) / (0.5 * (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, 9e+97], N[(N[(-2.0 / N[(N[(t - y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.99999999999999952e97

   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-/l* N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 89.7

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 8.99999999999999952e97 < z

   1. Initial program 81.9%

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

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\left(y - t\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{0.5 \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z\_m - t \cdot z\_m \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z\_m}{x} \cdot t}\\ \end{array} \end{array} \]

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 (<= (- (* y z_m) (* t z_m)) 2e+306)
    (* (/ -2.0 (* (- t y) z_m)) x)
    (/ -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 (((y * z_m) - (t * z_m)) <= 2e+306) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} 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 (((y * z_m) - (t * z_m)) <= 2d+306) then
        tmp = ((-2.0d0) / ((t - y) * z_m)) * x
    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 (((y * z_m) - (t * z_m)) <= 2e+306) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} 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 ((y * z_m) - (t * z_m)) <= 2e+306:
		tmp = (-2.0 / ((t - y) * z_m)) * x
	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 (Float64(Float64(y * z_m) - Float64(t * z_m)) <= 2e+306)
		tmp = Float64(Float64(-2.0 / Float64(Float64(t - y) * z_m)) * x);
	else
		tmp = Float64(-2.0 / Float64(Float64(z_m / 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 (((y * z_m) - (t * z_m)) <= 2e+306)
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	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[N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(-2.0 / N[(N[(t - y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(-2.0 / N[(N[(z$95$m / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 2.00000000000000003e306

   1. Initial program 92.6%

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

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-/l* N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 92.8

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 2.00000000000000003e306 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 43.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   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 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 69.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z}{x} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

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 9e+97)
    (* (/ -2.0 (* (- t y) z_m)) x)
    (* (/ 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 <= 9e+97) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} 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 <= 9d+97) then
        tmp = ((-2.0d0) / ((t - y) * z_m)) * x
    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 <= 9e+97) {
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	} 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 <= 9e+97:
		tmp = (-2.0 / ((t - y) * z_m)) * x
	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 <= 9e+97)
		tmp = Float64(Float64(-2.0 / Float64(Float64(t - y) * z_m)) * x);
	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 <= 9e+97)
		tmp = (-2.0 / ((t - y) * z_m)) * x;
	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, 9e+97], N[(N[(-2.0 / N[(N[(t - y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $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 < 8.99999999999999952e97

   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-/l* N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 89.7

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 8.99999999999999952e97 < z

   1. Initial program 81.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{-2}{\left(t - y\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{y \cdot z\_m} \cdot 2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 \cdot x}{\left(-t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -2.8e-26)
      t_1
      (if (<= y 1.2e-6) (/ (* 2.0 x) (* (- t) z_m)) 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 <= -2.8e-26) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = (2.0 * x) / (-t * z_m);
	} 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 <= (-2.8d-26)) then
        tmp = t_1
    else if (y <= 1.2d-6) then
        tmp = (2.0d0 * x) / (-t * z_m)
    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 <= -2.8e-26) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = (2.0 * x) / (-t * z_m);
	} 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 <= -2.8e-26:
		tmp = t_1
	elif y <= 1.2e-6:
		tmp = (2.0 * x) / (-t * z_m)
	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 <= -2.8e-26)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(-t) * z_m));
	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 <= -2.8e-26)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = (2.0 * x) / (-t * z_m);
	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, -2.8e-26], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(2.0 * x), $MachinePrecision] / N[((-t) * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8000000000000001e-26 or 1.1999999999999999e-6 < y

   1. Initial program 84.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   if -2.8000000000000001e-26 < y < 1.1999999999999999e-6

   1. Initial program 90.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 \cdot x}{\left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{y \cdot z\_m} \cdot 2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -2.8e-26) t_1 (if (<= y 1.2e-6) (* (/ 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 <= -2.8e-26) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		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 <= (-2.8d-26)) then
        tmp = t_1
    else if (y <= 1.2d-6) 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 <= -2.8e-26) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		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 <= -2.8e-26:
		tmp = t_1
	elif y <= 1.2e-6:
		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 <= -2.8e-26)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		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 <= -2.8e-26)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		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, -2.8e-26], t$95$1, If[LessEqual[y, 1.2e-6], 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 < -2.8000000000000001e-26 or 1.1999999999999999e-6 < y

   1. Initial program 84.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   if -2.8000000000000001e-26 < y < 1.1999999999999999e-6

   1. Initial program 90.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.2× speedup

Math

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

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 (* (- t y) 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 - y) * 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 - y) * 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 - y) * 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 - y) * 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(t - y) * 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 - y) * 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[(t - y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. associate-/l* N/A

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

   6. frac-2neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 88.2

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

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

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 90.5

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

4. Applied rewrites 90.5%

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

5. Final simplification 90.5%

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

Alternative 7: 92.2% accurate, 1.2× speedup

Math

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

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

Derivation

1. Initial program 88.0%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 90.3

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

4. Applied rewrites 90.3%

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

5. Final simplification 90.3%

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

Alternative 8: 53.3% 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 88.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
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 53.5

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

5. Applied rewrites 53.5%

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

Developer Target 1: 97.2% 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 2024270 
(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))))