Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.1% → 96.0%

Time: 12.6s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t - y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{0.5}{\frac{x\_m}{y - t}}}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2.7e+70)
    (* x_m (/ (/ -2.0 (- t y)) z))
    (/ (/ 1.0 (/ 0.5 (/ x_m (- y t)))) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2.7e+70) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (1.0 / (0.5 / (x_m / (y - t)))) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 2.7d+70) then
        tmp = x_m * (((-2.0d0) / (t - y)) / z)
    else
        tmp = (1.0d0 / (0.5d0 / (x_m / (y - t)))) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2.7e+70) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (1.0 / (0.5 / (x_m / (y - t)))) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 2.7e+70:
		tmp = x_m * ((-2.0 / (t - y)) / z)
	else:
		tmp = (1.0 / (0.5 / (x_m / (y - t)))) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 2.7e+70)
		tmp = Float64(x_m * Float64(Float64(-2.0 / Float64(t - y)) / z));
	else
		tmp = Float64(Float64(1.0 / Float64(0.5 / Float64(x_m / Float64(y - t)))) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 2.7e+70)
		tmp = x_m * ((-2.0 / (t - y)) / z);
	else
		tmp = (1.0 / (0.5 / (x_m / (y - t)))) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 2.7e+70], N[(x$95$m * N[(N[(-2.0 / N[(t - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(0.5 / N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2.7e70

   1. Initial program 95.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 96.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 96.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{\left(t - y\right) \cdot z}\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{t - y}}{z}\right), x\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{z}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + t\right)\right)}}{z}\right), x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)}\right), z\right), x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)}\right), z\right), x\right) \]

      15. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t - y}\right), z\right), x\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(t - y\right)\right), z\right), x\right) \]

      19. --lowering--.f64 96.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(t, y\right)\right), z\right), x\right) \]

   6. Applied egg-rr 96.3%

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

   if 2.7e70 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.2%

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t} \cdot x\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{y - t}\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), x\right), z\right) \]

      9. --lowering--.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), x\right), z\right) \]

   4. Applied egg-rr 97.4%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot x}{y - t}\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot 2}{y - t}\right), z\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{y - t}{x \cdot 2}}\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{y - t}{x \cdot 2}\right)\right), z\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{x \cdot 2}{y - t}}\right)\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{2 \cdot x}{y - t}}\right)\right), z\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{2 \cdot \frac{x}{y - t}}\right)\right), z\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2}}{\frac{x}{y - t}}\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2}\right), \left(\frac{x}{y - t}\right)\right)\right), z\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{x}{y - t}\right)\right)\right), z\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, \left(y - t\right)\right)\right)\right), z\right) \]

      12. --lowering--.f64 97.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right)\right)\right), z\right) \]

   6. Applied egg-rr 97.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{0.5}{\frac{x}{y - t}}}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{0.5}{\frac{x}{y - t}}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.7e-5)
    (/ (* x_m 2.0) (* y z))
    (if (<= y 3.5e-31) (* x_m (/ -2.0 (* t z))) (* x_m (/ (/ 2.0 y) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.7e-5) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 3.5e-31) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = x_m * ((2.0 / y) / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.7d-5)) then
        tmp = (x_m * 2.0d0) / (y * z)
    else if (y <= 3.5d-31) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = x_m * ((2.0d0 / y) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.7e-5) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 3.5e-31) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = x_m * ((2.0 / y) / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -3.7e-5:
		tmp = (x_m * 2.0) / (y * z)
	elif y <= 3.5e-31:
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = x_m * ((2.0 / y) / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -3.7e-5)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	elseif (y <= 3.5e-31)
		tmp = Float64(x_m * Float64(-2.0 / Float64(t * z)));
	else
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -3.7e-5)
		tmp = (x_m * 2.0) / (y * z);
	elseif (y <= 3.5e-31)
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = x_m * ((2.0 / y) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -3.7e-5], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-31], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.69999999999999981e-5

   1. Initial program 92.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(z \cdot \color{blue}{y}\right)\right) \]

      2. *-lowering-*.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 78.0%

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

   if -3.69999999999999981e-5 < y < 3.49999999999999985e-31

   1. Initial program 97.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 97.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot t\right)\right), x\right) \]

      3. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, t\right)\right), x\right) \]

   7. Simplified 85.7%

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

   if 3.49999999999999985e-31 < y

   1. Initial program 90.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 90.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{\left(t - y\right) \cdot z}\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{t - y}}{z}\right), x\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{z}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + t\right)\right)}}{z}\right), x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)}\right), z\right), x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)}\right), z\right), x\right) \]

      15. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t - y}\right), z\right), x\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(t - y\right)\right), z\right), x\right) \]

      19. --lowering--.f64 91.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(t, y\right)\right), z\right), x\right) \]

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{y}\right)}, z\right), x\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 77.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, y\right), z\right), x\right) \]

   9. Simplified 77.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-32}:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -2e-11)
    (* x_m (/ 2.0 (* y z)))
    (if (<= y 1.3e-32) (* x_m (/ -2.0 (* t z))) (* x_m (/ (/ 2.0 y) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2e-11) {
		tmp = x_m * (2.0 / (y * z));
	} else if (y <= 1.3e-32) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = x_m * ((2.0 / y) / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2d-11)) then
        tmp = x_m * (2.0d0 / (y * z))
    else if (y <= 1.3d-32) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = x_m * ((2.0d0 / y) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2e-11) {
		tmp = x_m * (2.0 / (y * z));
	} else if (y <= 1.3e-32) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = x_m * ((2.0 / y) / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -2e-11:
		tmp = x_m * (2.0 / (y * z))
	elif y <= 1.3e-32:
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = x_m * ((2.0 / y) / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -2e-11)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z)));
	elseif (y <= 1.3e-32)
		tmp = Float64(x_m * Float64(-2.0 / Float64(t * z)));
	else
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -2e-11)
		tmp = x_m * (2.0 / (y * z));
	elseif (y <= 1.3e-32)
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = x_m * ((2.0 / y) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -2e-11], N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-32], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999988e-11

   1. Initial program 92.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(z \cdot \color{blue}{y}\right)\right) \]

      2. *-lowering-*.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 78.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(z \cdot y\right)\right), x\right) \]

      5. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(z, y\right)\right), x\right) \]

   7. Applied egg-rr 77.9%

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

   if -1.99999999999999988e-11 < y < 1.2999999999999999e-32

   1. Initial program 97.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 97.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot t\right)\right), x\right) \]

      3. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, t\right)\right), x\right) \]

   7. Simplified 85.7%

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

   if 1.2999999999999999e-32 < y

   1. Initial program 90.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 90.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{\left(t - y\right) \cdot z}\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{t - y}}{z}\right), x\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{z}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + t\right)\right)}}{z}\right), x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)}\right), z\right), x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)}\right), z\right), x\right) \]

      15. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t - y}\right), z\right), x\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(t - y\right)\right), z\right), x\right) \]

      19. --lowering--.f64 91.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(t, y\right)\right), z\right), x\right) \]

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{y}\right)}, z\right), x\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 77.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, y\right), z\right), x\right) \]

   9. Simplified 77.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{2}{y \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-30}:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ 2.0 (* y z)))))
   (*
    x_s
    (if (<= y -6.5e-5) t_1 (if (<= y 3e-30) (* x_m (/ -2.0 (* t z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (y * z));
	double tmp;
	if (y <= -6.5e-5) {
		tmp = t_1;
	} else if (y <= 3e-30) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (2.0d0 / (y * z))
    if (y <= (-6.5d-5)) then
        tmp = t_1
    else if (y <= 3d-30) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (y * z));
	double tmp;
	if (y <= -6.5e-5) {
		tmp = t_1;
	} else if (y <= 3e-30) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (2.0 / (y * z))
	tmp = 0
	if y <= -6.5e-5:
		tmp = t_1
	elif y <= 3e-30:
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(2.0 / Float64(y * z)))
	tmp = 0.0
	if (y <= -6.5e-5)
		tmp = t_1;
	elseif (y <= 3e-30)
		tmp = Float64(x_m * Float64(-2.0 / Float64(t * z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (2.0 / (y * z));
	tmp = 0.0;
	if (y <= -6.5e-5)
		tmp = t_1;
	elseif (y <= 3e-30)
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -6.5e-5], t$95$1, If[LessEqual[y, 3e-30], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999943e-5 or 2.9999999999999999e-30 < y

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(z \cdot \color{blue}{y}\right)\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 77.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(z \cdot y\right)\right), x\right) \]

      5. *-lowering-*.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(z, y\right)\right), x\right) \]

   7. Applied egg-rr 77.1%

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

   if -6.49999999999999943e-5 < y < 2.9999999999999999e-30

   1. Initial program 97.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 97.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot t\right)\right), x\right) \]

      3. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, t\right)\right), x\right) \]

   7. Simplified 85.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t - y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{2}{y - t}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 2.7e+70)
    (* x_m (/ (/ -2.0 (- t y)) z))
    (/ (* x_m (/ 2.0 (- y t))) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2.7e+70) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (x_m * (2.0 / (y - t))) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 2.7d+70) then
        tmp = x_m * (((-2.0d0) / (t - y)) / z)
    else
        tmp = (x_m * (2.0d0 / (y - t))) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 2.7e+70) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (x_m * (2.0 / (y - t))) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 2.7e+70:
		tmp = x_m * ((-2.0 / (t - y)) / z)
	else:
		tmp = (x_m * (2.0 / (y - t))) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 2.7e+70)
		tmp = Float64(x_m * Float64(Float64(-2.0 / Float64(t - y)) / z));
	else
		tmp = Float64(Float64(x_m * Float64(2.0 / Float64(y - t))) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 2.7e+70)
		tmp = x_m * ((-2.0 / (t - y)) / z);
	else
		tmp = (x_m * (2.0 / (y - t))) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 2.7e+70], N[(x$95$m * N[(N[(-2.0 / N[(t - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2.7e70

   1. Initial program 95.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 96.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 96.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{\left(t - y\right) \cdot z}\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{t - y}}{z}\right), x\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{z}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + t\right)\right)}}{z}\right), x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)}\right), z\right), x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)}\right), z\right), x\right) \]

      15. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t - y}\right), z\right), x\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(t - y\right)\right), z\right), x\right) \]

      19. --lowering--.f64 96.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(t, y\right)\right), z\right), x\right) \]

   6. Applied egg-rr 96.3%

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

   if 2.7e70 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.2%

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t} \cdot x\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{y - t}\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), x\right), z\right) \]

      9. --lowering--.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), x\right), z\right) \]

   4. Applied egg-rr 97.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y - t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 2e-57)
    (* x_m (/ (/ -2.0 (- t y)) z))
    (* (/ 2.0 (- y t)) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2e-57) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2d-57) then
        tmp = x_m * (((-2.0d0) / (t - y)) / z)
    else
        tmp = (2.0d0 / (y - t)) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 2e-57) {
		tmp = x_m * ((-2.0 / (t - y)) / z);
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 2e-57:
		tmp = x_m * ((-2.0 / (t - y)) / z)
	else:
		tmp = (2.0 / (y - t)) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 2e-57)
		tmp = Float64(x_m * Float64(Float64(-2.0 / Float64(t - y)) / z));
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 2e-57)
		tmp = x_m * ((-2.0 / (t - y)) / z);
	else
		tmp = (2.0 / (y - t)) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 2e-57], N[(x$95$m * N[(N[(-2.0 / N[(t - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999991e-57

   1. Initial program 95.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 95.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 95.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{\left(t - y\right) \cdot z}\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{t - y}}{z}\right), x\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t - y\right)\right)}}{z}\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{z}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + t\right)\right)}}{z}\right), x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y + \left(\mathsf{neg}\left(t\right)\right)}}{z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y - t\right)\right)}\right), z\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)}\right), z\right), x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)}\right), z\right), x\right) \]

      15. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t + \left(\mathsf{neg}\left(y\right)\right)}\right), z\right), x\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{t - y}\right), z\right), x\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(t - y\right)\right), z\right), x\right) \]

      19. --lowering--.f64 95.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(t, y\right)\right), z\right), x\right) \]

   6. Applied egg-rr 95.6%

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

   if 1.99999999999999991e-57 < z

   1. Initial program 90.0%

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

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

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{2}{y - t}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{2}}{y - t}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(2, \color{blue}{\left(y - t\right)}\right)\right) \]

      6. --lowering--.f64 97.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   4. Applied egg-rr 97.1%

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

4. Final simplification 96.1%

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

Alternative 7: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+65}:\\ \;\;\;\;x\_m \cdot \frac{-2}{\left(t - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 5e+65)
    (* x_m (/ -2.0 (* (- t y) z)))
    (* (/ 2.0 (- y t)) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 5e+65) {
		tmp = x_m * (-2.0 / ((t - y) * z));
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5d+65) then
        tmp = x_m * ((-2.0d0) / ((t - y) * z))
    else
        tmp = (2.0d0 / (y - t)) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 5e+65) {
		tmp = x_m * (-2.0 / ((t - y) * z));
	} else {
		tmp = (2.0 / (y - t)) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 5e+65:
		tmp = x_m * (-2.0 / ((t - y) * z))
	else:
		tmp = (2.0 / (y - t)) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 5e+65)
		tmp = Float64(x_m * Float64(-2.0 / Float64(Float64(t - y) * z)));
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 5e+65)
		tmp = x_m * (-2.0 / ((t - y) * z));
	else
		tmp = (2.0 / (y - t)) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 5e+65], N[(x$95$m * N[(-2.0 / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999973e65

   1. Initial program 96.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

      16. --lowering--.f64 96.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

   4. Applied egg-rr 96.0%

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

   if 4.99999999999999973e65 < z

   1. Initial program 84.5%

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

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

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{2}{y - t}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{2}}{y - t}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(2, \color{blue}{\left(y - t\right)}\right)\right) \]

      6. --lowering--.f64 95.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   4. Applied egg-rr 95.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \frac{-2}{\left(t - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

   13. associate--r+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

   16. --lowering--.f64 94.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

4. Applied egg-rr 94.5%

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

5. Final simplification 94.5%

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

Alternative 9: 52.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)}\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(y \cdot z - t \cdot z\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(z \cdot \left(y - t\right)\right)\right)\right), x\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(y - t\right)\right)\right)\right)\right), x\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y - t\right)\right)\right)\right), x\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right), x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(0 - \left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right), x\right) \]

   13. associate--r+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - y\right)\right)\right), x\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - y\right)\right)\right), x\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \left(t - y\right)\right)\right), x\right) \]

   16. --lowering--.f64 94.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, y\right)\right)\right), x\right) \]

4. Applied egg-rr 94.5%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(z \cdot t\right)\right), x\right) \]

   3. *-lowering-*.f64 51.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(z, t\right)\right), x\right) \]

7. Simplified 51.6%

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

8. Final simplification 51.6%

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

Developer Target 1: 96.5% 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 2024161 
(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))))