Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.9% → 97.8%

Time: 9.7s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - z\_m \cdot t\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{2}{z\_m} \cdot x}{y - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y - t}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* y z_m) (* z_m t))))
   (*
    z_s
    (if (<= t_1 -5e+136)
      (/ (* (/ 2.0 z_m) x) (- y t))
      (if (<= t_1 5e+197)
        (/ x (* z_m (* (- y t) 0.5)))
        (* (/ 2.0 z_m) (/ x (- y t))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -5e+136) {
		tmp = ((2.0 / z_m) * x) / (y - t);
	} else if (t_1 <= 5e+197) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z_m) - (z_m * t)
    if (t_1 <= (-5d+136)) then
        tmp = ((2.0d0 / z_m) * x) / (y - t)
    else if (t_1 <= 5d+197) then
        tmp = x / (z_m * ((y - t) * 0.5d0))
    else
        tmp = (2.0d0 / z_m) * (x / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -5e+136) {
		tmp = ((2.0 / z_m) * x) / (y - t);
	} else if (t_1 <= 5e+197) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (y * z_m) - (z_m * t)
	tmp = 0
	if t_1 <= -5e+136:
		tmp = ((2.0 / z_m) * x) / (y - t)
	elif t_1 <= 5e+197:
		tmp = x / (z_m * ((y - t) * 0.5))
	else:
		tmp = (2.0 / z_m) * (x / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(z_m * t))
	tmp = 0.0
	if (t_1 <= -5e+136)
		tmp = Float64(Float64(Float64(2.0 / z_m) * x) / Float64(y - t));
	elseif (t_1 <= 5e+197)
		tmp = Float64(x / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (y * z_m) - (z_m * t);
	tmp = 0.0;
	if (t_1 <= -5e+136)
		tmp = ((2.0 / z_m) * x) / (y - t);
	elseif (t_1 <= 5e+197)
		tmp = x / (z_m * ((y - t) * 0.5));
	else
		tmp = (2.0 / z_m) * (x / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$1, -5e+136], N[(N[(N[(2.0 / z$95$m), $MachinePrecision] * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+197], N[(x / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -5.0000000000000002e136

   1. Initial program 85.5%

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

      1. *-commutative N/A

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

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

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if -5.0000000000000002e136 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000009e197

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval 98.3

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

   4. Applied egg-rr 98.3%

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

   if 5.00000000000000009e197 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 72.5%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - z\_m \cdot t\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y - t}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* y z_m) (* z_m t))))
   (*
    z_s
    (if (<= t_1 -2e+169)
      (* (/ x z_m) (/ 2.0 (- y t)))
      (if (<= t_1 5e+197)
        (/ x (* z_m (* (- y t) 0.5)))
        (* (/ 2.0 z_m) (/ x (- y t))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else if (t_1 <= 5e+197) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z_m) - (z_m * t)
    if (t_1 <= (-2d+169)) then
        tmp = (x / z_m) * (2.0d0 / (y - t))
    else if (t_1 <= 5d+197) then
        tmp = x / (z_m * ((y - t) * 0.5d0))
    else
        tmp = (2.0d0 / z_m) * (x / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else if (t_1 <= 5e+197) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (y * z_m) - (z_m * t)
	tmp = 0
	if t_1 <= -2e+169:
		tmp = (x / z_m) * (2.0 / (y - t))
	elif t_1 <= 5e+197:
		tmp = x / (z_m * ((y - t) * 0.5))
	else:
		tmp = (2.0 / z_m) * (x / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(z_m * t))
	tmp = 0.0
	if (t_1 <= -2e+169)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	elseif (t_1 <= 5e+197)
		tmp = Float64(x / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (y * z_m) - (z_m * t);
	tmp = 0.0;
	if (t_1 <= -2e+169)
		tmp = (x / z_m) * (2.0 / (y - t));
	elseif (t_1 <= 5e+197)
		tmp = x / (z_m * ((y - t) * 0.5));
	else
		tmp = (2.0 / z_m) * (x / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$1, -2e+169], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+197], N[(x / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.99999999999999987e169

   1. Initial program 83.8%

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

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if -1.99999999999999987e169 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000009e197

   1. Initial program 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval 98.3

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

   4. Applied egg-rr 98.3%

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

   if 5.00000000000000009e197 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 72.5%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{z\_m} \cdot \frac{2}{y - t}\\ t_2 := y \cdot z\_m - z\_m \cdot t\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x z_m) (/ 2.0 (- y t)))) (t_2 (- (* y z_m) (* z_m t))))
   (*
    z_s
    (if (<= t_2 -2e+169)
      t_1
      (if (<= t_2 4e+306) (/ x (* z_m (* (- y t) 0.5))) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / z_m) * (2.0 / (y - t));
	double t_2 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_2 <= -2e+169) {
		tmp = t_1;
	} else if (t_2 <= 4e+306) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / z_m) * (2.0d0 / (y - t))
    t_2 = (y * z_m) - (z_m * t)
    if (t_2 <= (-2d+169)) then
        tmp = t_1
    else if (t_2 <= 4d+306) then
        tmp = x / (z_m * ((y - t) * 0.5d0))
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / z_m) * (2.0 / (y - t));
	double t_2 = (y * z_m) - (z_m * t);
	double tmp;
	if (t_2 <= -2e+169) {
		tmp = t_1;
	} else if (t_2 <= 4e+306) {
		tmp = x / (z_m * ((y - t) * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / z_m) * (2.0 / (y - t))
	t_2 = (y * z_m) - (z_m * t)
	tmp = 0
	if t_2 <= -2e+169:
		tmp = t_1
	elif t_2 <= 4e+306:
		tmp = x / (z_m * ((y - t) * 0.5))
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)))
	t_2 = Float64(Float64(y * z_m) - Float64(z_m * t))
	tmp = 0.0
	if (t_2 <= -2e+169)
		tmp = t_1;
	elseif (t_2 <= 4e+306)
		tmp = Float64(x / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / z_m) * (2.0 / (y - t));
	t_2 = (y * z_m) - (z_m * t);
	tmp = 0.0;
	if (t_2 <= -2e+169)
		tmp = t_1;
	elseif (t_2 <= 4e+306)
		tmp = x / (z_m * ((y - t) * 0.5));
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$2, -2e+169], t$95$1, If[LessEqual[t$95$2, 4e+306], N[(x / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.99999999999999987e169 or 4.00000000000000007e306 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 75.6%

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

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if -1.99999999999999987e169 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.00000000000000007e306

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval 98.4

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x \cdot -2}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(y \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x -2.0) (* z_m t))))
   (*
    z_s
    (if (<= t -2.9e+25) t_1 (if (<= t 7.2e-60) (/ x (* z_m (* y 0.5))) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x * -2.0) / (z_m * t);
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x / (z_m * (y * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * (-2.0d0)) / (z_m * t)
    if (t <= (-2.9d+25)) then
        tmp = t_1
    else if (t <= 7.2d-60) then
        tmp = x / (z_m * (y * 0.5d0))
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x * -2.0) / (z_m * t);
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x / (z_m * (y * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x * -2.0) / (z_m * t)
	tmp = 0
	if t <= -2.9e+25:
		tmp = t_1
	elif t <= 7.2e-60:
		tmp = x / (z_m * (y * 0.5))
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x * -2.0) / Float64(z_m * t))
	tmp = 0.0
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = Float64(x / Float64(z_m * Float64(y * 0.5)));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x * -2.0) / (z_m * t);
	tmp = 0.0;
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = x / (z_m * (y * 0.5));
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x * -2.0), $MachinePrecision] / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -2.9e+25], t$95$1, If[LessEqual[t, 7.2e-60], N[(x / N[(z$95$m * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999999e25 or 7.2e-60 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

   if -2.8999999999999999e25 < t < 7.2e-60

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval 94.9

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.5

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

   7. Simplified 76.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := x \cdot \frac{-2}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(y \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* x (/ -2.0 (* z_m t)))))
   (*
    z_s
    (if (<= t -2.9e+25) t_1 (if (<= t 7.2e-60) (/ x (* z_m (* y 0.5))) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = x * (-2.0 / (z_m * t));
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x / (z_m * (y * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((-2.0d0) / (z_m * t))
    if (t <= (-2.9d+25)) then
        tmp = t_1
    else if (t <= 7.2d-60) then
        tmp = x / (z_m * (y * 0.5d0))
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = x * (-2.0 / (z_m * t));
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x / (z_m * (y * 0.5));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = x * (-2.0 / (z_m * t))
	tmp = 0
	if t <= -2.9e+25:
		tmp = t_1
	elif t <= 7.2e-60:
		tmp = x / (z_m * (y * 0.5))
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(x * Float64(-2.0 / Float64(z_m * t)))
	tmp = 0.0
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = Float64(x / Float64(z_m * Float64(y * 0.5)));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = x * (-2.0 / (z_m * t));
	tmp = 0.0;
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = x / (z_m * (y * 0.5));
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -2.9e+25], t$95$1, If[LessEqual[t, 7.2e-60], N[(x / N[(z$95$m * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999999e25 or 7.2e-60 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -2.8999999999999999e25 < t < 7.2e-60

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval 94.9

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.5

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

   7. Simplified 76.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := x \cdot \frac{-2}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* x (/ -2.0 (* z_m t)))))
   (*
    z_s
    (if (<= t -2.9e+25) t_1 (if (<= t 7.2e-60) (* x (/ 2.0 (* y z_m))) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = x * (-2.0 / (z_m * t));
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x * (2.0 / (y * z_m));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((-2.0d0) / (z_m * t))
    if (t <= (-2.9d+25)) then
        tmp = t_1
    else if (t <= 7.2d-60) then
        tmp = x * (2.0d0 / (y * z_m))
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = x * (-2.0 / (z_m * t));
	double tmp;
	if (t <= -2.9e+25) {
		tmp = t_1;
	} else if (t <= 7.2e-60) {
		tmp = x * (2.0 / (y * z_m));
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = x * (-2.0 / (z_m * t))
	tmp = 0
	if t <= -2.9e+25:
		tmp = t_1
	elif t <= 7.2e-60:
		tmp = x * (2.0 / (y * z_m))
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(x * Float64(-2.0 / Float64(z_m * t)))
	tmp = 0.0
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = Float64(x * Float64(2.0 / Float64(y * z_m)));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = x * (-2.0 / (z_m * t));
	tmp = 0.0;
	if (t <= -2.9e+25)
		tmp = t_1;
	elseif (t <= 7.2e-60)
		tmp = x * (2.0 / (y * z_m));
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -2.9e+25], t$95$1, If[LessEqual[t, 7.2e-60], N[(x * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999999e25 or 7.2e-60 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -2.8999999999999999e25 < t < 7.2e-60

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.8

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.2

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

   7. Simplified 75.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower--.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval 93.5

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

4. Applied egg-rr 93.5%

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

Alternative 8: 91.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 90.5

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 92.9

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

4. Applied egg-rr 92.9%

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

5. Final simplification 92.9%

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

Alternative 9: 53.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 56.1

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

5. Simplified 56.1%

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 56.1

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 56.1

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

7. Applied egg-rr 56.1%

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

8. Final simplification 56.1%

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

Developer Target 1: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ t_2 := \frac{x \cdot 2}{y \cdot z - t \cdot z}\\ \mathbf{if}\;t\_2 < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (* (- y t) z)) 2.0))
        (t_2 (/ (* x 2.0) (- (* y z) (* t z)))))
   (if (< t_2 -2.559141628295061e-13)
     t_1
     (if (< t_2 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / ((y - t) * z)) * 2.0d0
    t_2 = (x * 2.0d0) / ((y * z) - (t * z))
    if (t_2 < (-2.559141628295061d-13)) then
        tmp = t_1
    else if (t_2 < 1.045027827330126d-269) then
        tmp = ((x / z) * 2.0d0) / (y - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / ((y - t) * z)) * 2.0
	t_2 = (x * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if t_2 < -2.559141628295061e-13:
		tmp = t_1
	elif t_2 < 1.045027827330126e-269:
		tmp = ((x / z) * 2.0) / (y - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0)
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = Float64(Float64(Float64(x / z) * 2.0) / Float64(y - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / ((y - t) * z)) * 2.0;
	t_2 = (x * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = ((x / z) * 2.0) / (y - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -2.559141628295061e-13], t$95$1, If[Less[t$95$2, 1.045027827330126e-269], N[(N[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024207 
(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))))