Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 97.8%

Time: 7.9s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+195} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{\frac{x}{z\_m}}{\left(y - t\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \end{array} \end{array} \end{array} \]

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) (* t z_m))))
   (*
    z_s
    (if (or (<= t_1 -5e+195) (not (<= t_1 2e+166)))
      (/ (/ x z_m) (* (- y t) 0.5))
      (/ (* x 2.0) (* (- y t) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166)) {
		tmp = (x / z_m) / ((y - t) * 0.5);
	} else {
		tmp = (x * 2.0) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z_m) - (t * z_m)
    if ((t_1 <= (-5d+195)) .or. (.not. (t_1 <= 2d+166))) then
        tmp = (x / z_m) / ((y - t) * 0.5d0)
    else
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166)) {
		tmp = (x / z_m) / ((y - t) * 0.5);
	} else {
		tmp = (x * 2.0) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (y * z_m) - (t * z_m)
	tmp = 0
	if (t_1 <= -5e+195) or not (t_1 <= 2e+166):
		tmp = (x / z_m) / ((y - t) * 0.5)
	else:
		tmp = (x * 2.0) / ((y - t) * z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166))
		tmp = Float64(Float64(x / z_m) / Float64(Float64(y - t) * 0.5));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if ((t_1 <= -5e+195) || ~((t_1 <= 2e+166)))
		tmp = (x / z_m) / ((y - t) * 0.5);
	else
		tmp = (x * 2.0) / ((y - t) * z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[Or[LessEqual[t$95$1, -5e+195], N[Not[LessEqual[t$95$1, 2e+166]], $MachinePrecision]], N[(N[(x / z$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.9999999999999998e195 or 1.99999999999999988e166 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 75.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if -4.9999999999999998e195 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999988e166

   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

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 97.7

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

   4. Applied rewrites 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \leq -5 \cdot 10^{+195} \lor \neg \left(y \cdot z - t \cdot z \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\left(y - t\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+195} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \end{array} \end{array} \end{array} \]

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) (* t z_m))))
   (*
    z_s
    (if (or (<= t_1 -5e+195) (not (<= t_1 2e+166)))
      (* (/ x z_m) (/ 2.0 (- y t)))
      (/ (* x 2.0) (* (- y t) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166)) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x * 2.0) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z_m) - (t * z_m)
    if ((t_1 <= (-5d+195)) .or. (.not. (t_1 <= 2d+166))) then
        tmp = (x / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166)) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x * 2.0) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (y * z_m) - (t * z_m)
	tmp = 0
	if (t_1 <= -5e+195) or not (t_1 <= 2e+166):
		tmp = (x / z_m) * (2.0 / (y - t))
	else:
		tmp = (x * 2.0) / ((y - t) * z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if ((t_1 <= -5e+195) || !(t_1 <= 2e+166))
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if ((t_1 <= -5e+195) || ~((t_1 <= 2e+166)))
		tmp = (x / z_m) * (2.0 / (y - t));
	else
		tmp = (x * 2.0) / ((y - t) * z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[Or[LessEqual[t$95$1, -5e+195], N[Not[LessEqual[t$95$1, 2e+166]], $MachinePrecision]], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.9999999999999998e195 or 1.99999999999999988e166 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -4.9999999999999998e195 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999988e166

   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

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 97.7

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

   4. Applied rewrites 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \leq -5 \cdot 10^{+195} \lor \neg \left(y \cdot z - t \cdot z \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{x \cdot 2}{\mathsf{fma}\left(-z\_m, t, z\_m \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m}}{\frac{y - t}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (* x 2.0) 5e-41)
    (/ (* x 2.0) (fma (- z_m) t (* z_m y)))
    (/ (/ 2.0 z_m) (/ (- y t) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 5e-41) {
		tmp = (x * 2.0) / fma(-z_m, t, (z_m * y));
	} else {
		tmp = (2.0 / z_m) / ((y - t) / x);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 5e-41)
		tmp = Float64(Float64(x * 2.0) / fma(Float64(-z_m), t, Float64(z_m * y)));
	else
		tmp = Float64(Float64(2.0 / z_m) / Float64(Float64(y - t) / x));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 5e-41], N[(N[(x * 2.0), $MachinePrecision] / N[((-z$95$m) * t + N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.9999999999999996e-41

   1. Initial program 91.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   if 4.9999999999999996e-41 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 4: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z\_m - t \cdot z\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{-0.5 \cdot t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* y z_m) (* t z_m)) 2e+307)
    (/ (* x 2.0) (* (- y t) z_m))
    (/ (/ x z_m) (* -0.5 t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= 2e+307) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = (x / z_m) / (-0.5 * t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((y * z_m) - (t * z_m)) <= 2d+307) then
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    else
        tmp = (x / z_m) / ((-0.5d0) * t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= 2e+307) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = (x / z_m) / (-0.5 * t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if ((y * z_m) - (t * z_m)) <= 2e+307:
		tmp = (x * 2.0) / ((y - t) * z_m)
	else:
		tmp = (x / z_m) / (-0.5 * t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(y * z_m) - Float64(t * z_m)) <= 2e+307)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(x / z_m) / Float64(-0.5 * t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (((y * z_m) - (t * z_m)) <= 2e+307)
		tmp = (x * 2.0) / ((y - t) * z_m);
	else
		tmp = (x / z_m) / (-0.5 * t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(-0.5 * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999997e307

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 94.0

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

   4. Applied rewrites 94.0%

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

   if 1.99999999999999997e307 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 50.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{-1}{2} \cdot t}} \]
   6. Step-by-step derivation

      1. lower-*.f64 62.3

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{-0.5 \cdot t}} \]

   7. Applied rewrites 62.3%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{-0.5 \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z\_m - t \cdot z\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* y z_m) (* t z_m)) 2e+307)
    (/ (* x 2.0) (* (- y t) z_m))
    (* (/ x z_m) (/ -2.0 t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= 2e+307) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((y * z_m) - (t * z_m)) <= 2d+307) then
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    else
        tmp = (x / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= 2e+307) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if ((y * z_m) - (t * z_m)) <= 2e+307:
		tmp = (x * 2.0) / ((y - t) * z_m)
	else:
		tmp = (x / z_m) * (-2.0 / t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(y * z_m) - Float64(t * z_m)) <= 2e+307)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (((y * z_m) - (t * z_m)) <= 2e+307)
		tmp = (x * 2.0) / ((y - t) * z_m);
	else
		tmp = (x / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999997e307

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 94.0

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

   4. Applied rewrites 94.0%

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

   if 1.99999999999999997e307 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 50.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 62.3

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

   7. Applied rewrites 62.3%

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

Alternative 6: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{x \cdot 2}{\mathsf{fma}\left(-z\_m, t, z\_m \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (* x 2.0) 2e-51)
    (/ (* x 2.0) (fma (- z_m) t (* z_m y)))
    (/ (/ (* 2.0 x) (- y t)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 2e-51) {
		tmp = (x * 2.0) / fma(-z_m, t, (z_m * y));
	} else {
		tmp = ((2.0 * x) / (y - t)) / z_m;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 2e-51)
		tmp = Float64(Float64(x * 2.0) / fma(Float64(-z_m), t, Float64(z_m * y)));
	else
		tmp = Float64(Float64(Float64(2.0 * x) / Float64(y - t)) / z_m);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 2e-51], N[(N[(x * 2.0), $MachinePrecision] / N[((-z$95$m) * t + N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2e-51

   1. Initial program 91.1%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   if 2e-51 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 7: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-65} \lor \neg \left(t \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot 2}{\left(-t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -4.8e-65) (not (<= t 8.5e-9)))
    (/ (* x 2.0) (* (- t) z_m))
    (* (/ x (* z_m y)) 2.0))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -4.8e-65) || !(t <= 8.5e-9)) {
		tmp = (x * 2.0) / (-t * z_m);
	} else {
		tmp = (x / (z_m * y)) * 2.0;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.8d-65)) .or. (.not. (t <= 8.5d-9))) then
        tmp = (x * 2.0d0) / (-t * z_m)
    else
        tmp = (x / (z_m * y)) * 2.0d0
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -4.8e-65) || !(t <= 8.5e-9)) {
		tmp = (x * 2.0) / (-t * z_m);
	} else {
		tmp = (x / (z_m * y)) * 2.0;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (t <= -4.8e-65) or not (t <= 8.5e-9):
		tmp = (x * 2.0) / (-t * z_m)
	else:
		tmp = (x / (z_m * y)) * 2.0
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((t <= -4.8e-65) || !(t <= 8.5e-9))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(-t) * z_m));
	else
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((t <= -4.8e-65) || ~((t <= 8.5e-9)))
		tmp = (x * 2.0) / (-t * z_m);
	else
		tmp = (x / (z_m * y)) * 2.0;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[t, -4.8e-65], N[Not[LessEqual[t, 8.5e-9]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[((-t) * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -4.8000000000000003e-65 or 8.5e-9 < t

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -4.8000000000000003e-65 < t < 8.5e-9

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-65} \lor \neg \left(t \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot 2}{\left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-65} \lor \neg \left(t \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -4.8e-65) (not (<= t 8.5e-9)))
    (* (/ x (* t z_m)) -2.0)
    (* (/ x (* z_m y)) 2.0))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -4.8e-65) || !(t <= 8.5e-9)) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = (x / (z_m * y)) * 2.0;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.8d-65)) .or. (.not. (t <= 8.5d-9))) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else
        tmp = (x / (z_m * y)) * 2.0d0
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -4.8e-65) || !(t <= 8.5e-9)) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = (x / (z_m * y)) * 2.0;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (t <= -4.8e-65) or not (t <= 8.5e-9):
		tmp = (x / (t * z_m)) * -2.0
	else:
		tmp = (x / (z_m * y)) * 2.0
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((t <= -4.8e-65) || !(t <= 8.5e-9))
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	else
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((t <= -4.8e-65) || ~((t <= 8.5e-9)))
		tmp = (x / (t * z_m)) * -2.0;
	else
		tmp = (x / (z_m * y)) * 2.0;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[t, -4.8e-65], N[Not[LessEqual[t, 8.5e-9]], $MachinePrecision]], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -4.8000000000000003e-65 or 8.5e-9 < t

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -4.8000000000000003e-65 < t < 8.5e-9

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-65} \lor \neg \left(t \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 90.9

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

4. Applied rewrites 90.9%

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

Alternative 10: 52.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 52.3

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

5. Applied rewrites 52.3%

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

Alternative 11: 52.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 52.3

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

5. Applied rewrites 52.3%

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

7. Applied rewrites 52.0%

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

Developer Target 1: 96.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(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))))