Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.2% → 98.0%

Time: 8.6s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{x}{z\_m}}{0.5 \cdot \left(y - t\right)}\\ t_2 := z\_m \cdot y - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+269}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (/ x z_m) (* 0.5 (- y t)))) (t_2 (- (* z_m y) (* t z_m))))
   (*
    z_s
    (if (<= t_2 -4e+269)
      t_1
      (if (<= t_2 5e+188) (/ (* 2.0 x) (* (- y t) z_m)) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / z_m) / (0.5 * (y - t));
	double t_2 = (z_m * y) - (t * z_m);
	double tmp;
	if (t_2 <= -4e+269) {
		tmp = t_1;
	} else if (t_2 <= 5e+188) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / z_m) / (0.5d0 * (y - t))
    t_2 = (z_m * y) - (t * z_m)
    if (t_2 <= (-4d+269)) then
        tmp = t_1
    else if (t_2 <= 5d+188) then
        tmp = (2.0d0 * x) / ((y - t) * z_m)
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / z_m) / (0.5 * (y - t));
	double t_2 = (z_m * y) - (t * z_m);
	double tmp;
	if (t_2 <= -4e+269) {
		tmp = t_1;
	} else if (t_2 <= 5e+188) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / z_m) / (0.5 * (y - t))
	t_2 = (z_m * y) - (t * z_m)
	tmp = 0
	if t_2 <= -4e+269:
		tmp = t_1
	elif t_2 <= 5e+188:
		tmp = (2.0 * x) / ((y - t) * z_m)
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / z_m) / Float64(0.5 * Float64(y - t)))
	t_2 = Float64(Float64(z_m * y) - Float64(t * z_m))
	tmp = 0.0
	if (t_2 <= -4e+269)
		tmp = t_1;
	elseif (t_2 <= 5e+188)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / z_m) / (0.5 * (y - t));
	t_2 = (z_m * y) - (t * z_m);
	tmp = 0.0;
	if (t_2 <= -4e+269)
		tmp = t_1;
	elseif (t_2 <= 5e+188)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / z$95$m), $MachinePrecision] / N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z$95$m * y), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$2, -4e+269], t$95$1, If[LessEqual[t$95$2, 5e+188], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.0000000000000002e269 or 5.0000000000000001e188 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 67.8%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 98.6

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

   4. Applied rewrites 98.6%

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

   if -4.0000000000000002e269 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000001e188

   1. Initial program 95.5%

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

      1. lift--.f64 N/A

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

      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 96.6

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

   4. Applied rewrites 96.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - t \cdot z \leq -4 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{x}{z}}{0.5 \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \cdot y - t \cdot z \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{0.5 \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t} \cdot \frac{x}{z\_m}\\ t_2 := z\_m \cdot y - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+269}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ 2.0 (- y t)) (/ x z_m))) (t_2 (- (* z_m y) (* t z_m))))
   (*
    z_s
    (if (<= t_2 -4e+269)
      t_1
      (if (<= t_2 4e+233) (/ (* 2.0 x) (* (- y t) z_m)) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (2.0 / (y - t)) * (x / z_m);
	double t_2 = (z_m * y) - (t * z_m);
	double tmp;
	if (t_2 <= -4e+269) {
		tmp = t_1;
	} else if (t_2 <= 4e+233) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 / (y - t)) * (x / z_m)
    t_2 = (z_m * y) - (t * z_m)
    if (t_2 <= (-4d+269)) then
        tmp = t_1
    else if (t_2 <= 4d+233) then
        tmp = (2.0d0 * x) / ((y - t) * z_m)
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (2.0 / (y - t)) * (x / z_m);
	double t_2 = (z_m * y) - (t * z_m);
	double tmp;
	if (t_2 <= -4e+269) {
		tmp = t_1;
	} else if (t_2 <= 4e+233) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (2.0 / (y - t)) * (x / z_m)
	t_2 = (z_m * y) - (t * z_m)
	tmp = 0
	if t_2 <= -4e+269:
		tmp = t_1
	elif t_2 <= 4e+233:
		tmp = (2.0 * x) / ((y - t) * z_m)
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(2.0 / Float64(y - t)) * Float64(x / z_m))
	t_2 = Float64(Float64(z_m * y) - Float64(t * z_m))
	tmp = 0.0
	if (t_2 <= -4e+269)
		tmp = t_1;
	elseif (t_2 <= 4e+233)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (2.0 / (y - t)) * (x / z_m);
	t_2 = (z_m * y) - (t * z_m);
	tmp = 0.0;
	if (t_2 <= -4e+269)
		tmp = t_1;
	elseif (t_2 <= 4e+233)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z$95$m * y), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$2, -4e+269], t$95$1, If[LessEqual[t$95$2, 4e+233], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.0000000000000002e269 or 3.99999999999999989e233 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 -4.0000000000000002e269 < (-.f64 (*.f64 y z) (*.f64 t z)) < 3.99999999999999989e233

   1. Initial program 95.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - t \cdot z \leq -4 \cdot 10^{+269}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot y - t \cdot z \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2 \cdot x}{z\_m \cdot y - t \cdot z\_m} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

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 (<= (/ (* 2.0 x) (- (* z_m y) (* t z_m))) -5e-324)
    (/ (* 2.0 x) (* (- y t) z_m))
    (/ (/ (* 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 (((2.0 * x) / ((z_m * y) - (t * z_m))) <= -5e-324) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 * x) / (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) :: tmp
    if (((2.0d0 * x) / ((z_m * y) - (t * z_m))) <= (-5d-324)) then
        tmp = (2.0d0 * x) / ((y - t) * z_m)
    else
        tmp = ((2.0d0 * x) / (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 tmp;
	if (((2.0 * x) / ((z_m * y) - (t * z_m))) <= -5e-324) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 * x) / (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):
	tmp = 0
	if ((2.0 * x) / ((z_m * y) - (t * z_m))) <= -5e-324:
		tmp = (2.0 * x) / ((y - t) * z_m)
	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(Float64(2.0 * x) / Float64(Float64(z_m * y) - Float64(t * z_m))) <= -5e-324)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(2.0 * x) / 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)
	tmp = 0.0;
	if (((2.0 * x) / ((z_m * y) - (t * z_m))) <= -5e-324)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = ((2.0 * x) / (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_] := N[(z$95$s * If[LessEqual[N[(N[(2.0 * x), $MachinePrecision] / N[(N[(z$95$m * y), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-324], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $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 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.94066e-324

   1. Initial program 95.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 96.3

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

   4. Applied rewrites 96.3%

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

   if -4.94066e-324 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot x}{z \cdot y - t \cdot z} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{z\_m \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* t z_m)) -2.0)))
   (*
    z_s
    (if (<= t -1e-113) t_1 (if (<= t 2.4e+28) (* (/ 2.0 (* z_m y)) x) 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 / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1e-113) {
		tmp = t_1;
	} else if (t <= 2.4e+28) {
		tmp = (2.0 / (z_m * y)) * x;
	} 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 / (t * z_m)) * (-2.0d0)
    if (t <= (-1d-113)) then
        tmp = t_1
    else if (t <= 2.4d+28) then
        tmp = (2.0d0 / (z_m * y)) * x
    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 / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1e-113) {
		tmp = t_1;
	} else if (t <= 2.4e+28) {
		tmp = (2.0 / (z_m * y)) * x;
	} 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 / (t * z_m)) * -2.0
	tmp = 0
	if t <= -1e-113:
		tmp = t_1
	elif t <= 2.4e+28:
		tmp = (2.0 / (z_m * y)) * x
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -1e-113)
		tmp = t_1;
	elseif (t <= 2.4e+28)
		tmp = Float64(Float64(2.0 / Float64(z_m * y)) * x);
	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 / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -1e-113)
		tmp = t_1;
	elseif (t <= 2.4e+28)
		tmp = (2.0 / (z_m * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -1e-113], t$95$1, If[LessEqual[t, 2.4e+28], N[(N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.99999999999999979e-114 or 2.39999999999999981e28 < t

   1. Initial program 83.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   if -9.99999999999999979e-114 < t < 2.39999999999999981e28

   1. Initial program 93.3%

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

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

      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 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.6

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

   9. Applied rewrites 82.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{z \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x (* t z_m)) -2.0)))
   (*
    z_s
    (if (<= t -1e-113) t_1 (if (<= t 4.4e-28) (* (/ x (* z_m y)) 2.0) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1e-113) {
		tmp = t_1;
	} else if (t <= 4.4e-28) {
		tmp = (x / (z_m * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / (t * z_m)) * (-2.0d0)
    if (t <= (-1d-113)) then
        tmp = t_1
    else if (t <= 4.4d-28) then
        tmp = (x / (z_m * y)) * 2.0d0
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1e-113) {
		tmp = t_1;
	} else if (t <= 4.4e-28) {
		tmp = (x / (z_m * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / (t * z_m)) * -2.0
	tmp = 0
	if t <= -1e-113:
		tmp = t_1
	elif t <= 4.4e-28:
		tmp = (x / (z_m * y)) * 2.0
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -1e-113)
		tmp = t_1;
	elseif (t <= 4.4e-28)
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -1e-113)
		tmp = t_1;
	elseif (t <= 4.4e-28)
		tmp = (x / (z_m * y)) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -1e-113], t$95$1, If[LessEqual[t, 4.4e-28], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.99999999999999979e-114 or 4.39999999999999992e-28 < t

   1. Initial program 83.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 -9.99999999999999979e-114 < t < 4.39999999999999992e-28

   1. Initial program 94.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 6: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+257}:\\ \;\;\;\;\frac{-2}{\frac{z\_m}{x} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1.4e+257)
    (/ -2.0 (* (/ z_m x) t))
    (/ (* 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 (t <= -1.4e+257) {
		tmp = -2.0 / ((z_m / x) * t);
	} else {
		tmp = (2.0 * x) / ((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) :: tmp
    if (t <= (-1.4d+257)) then
        tmp = (-2.0d0) / ((z_m / x) * t)
    else
        tmp = (2.0d0 * x) / ((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 tmp;
	if (t <= -1.4e+257) {
		tmp = -2.0 / ((z_m / x) * t);
	} else {
		tmp = (2.0 * x) / ((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):
	tmp = 0
	if t <= -1.4e+257:
		tmp = -2.0 / ((z_m / x) * t)
	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 (t <= -1.4e+257)
		tmp = Float64(-2.0 / Float64(Float64(z_m / x) * t));
	else
		tmp = Float64(Float64(2.0 * x) / 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)
	tmp = 0.0;
	if (t <= -1.4e+257)
		tmp = -2.0 / ((z_m / x) * t);
	else
		tmp = (2.0 * x) / ((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_] := N[(z$95$s * If[LessEqual[t, -1.4e+257], N[(-2.0 / N[(N[(z$95$m / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -1.3999999999999999e257

   1. Initial program 63.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 98.1%

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

   if -1.3999999999999999e257 < t

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 91.6

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

   4. Applied rewrites 91.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+257}:\\ \;\;\;\;\frac{-2}{\frac{z}{x} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. lift--.f64 N/A

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

   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 90.2

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

4. Applied rewrites 90.2%

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

5. Final simplification 90.2%

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

Alternative 8: 91.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. associate-/l* N/A

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

   6. frac-2neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 87.4

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

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

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 90.2

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

4. Applied rewrites 90.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift--.f64 N/A

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

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

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

   12. frac-2neg N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

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

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

   17. lift--.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. lower-/.f64 90.2

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 90.2

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

6. Applied rewrites 90.2%

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

7. Final simplification 90.2%

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

Alternative 9: 53.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 55.5

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

5. Applied rewrites 55.5%

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

Alternative 10: 53.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 55.5

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

5. Applied rewrites 55.5%

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

7. Applied rewrites 55.4%

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

8. Final simplification 55.4%

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

Developer Target 1: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024284 
(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))))