Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.7% → 98.2%

Time: 10.5s

Alternatives: 12

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: 89.7% 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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := z\_m \cdot y - z\_m \cdot t\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+191}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{y - t}{x}}}{z\_m}\\ \end{array} \end{array} \end{array} \]

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 (- (* z_m y) (* z_m t))))
   (*
    z_s
    (if (<= t_1 -4e+191)
      (* (/ x (- y t)) (/ 2.0 z_m))
      (if (<= t_1 2e+220)
        (/ (* x 2.0) (* z_m (- y t)))
        (/ (/ 2.0 (/ (- y t) x)) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (z_m * y) - (z_m * t);
	double tmp;
	if (t_1 <= -4e+191) {
		tmp = (x / (y - t)) * (2.0 / z_m);
	} else if (t_1 <= 2e+220) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / ((y - t) / x)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * y) - (z_m * t)
    if (t_1 <= (-4d+191)) then
        tmp = (x / (y - t)) * (2.0d0 / z_m)
    else if (t_1 <= 2d+220) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (2.0d0 / ((y - t) / x)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (z_m * y) - (z_m * t);
	double tmp;
	if (t_1 <= -4e+191) {
		tmp = (x / (y - t)) * (2.0 / z_m);
	} else if (t_1 <= 2e+220) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / ((y - t) / x)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (z_m * y) - (z_m * t)
	tmp = 0
	if t_1 <= -4e+191:
		tmp = (x / (y - t)) * (2.0 / z_m)
	elif t_1 <= 2e+220:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (2.0 / ((y - t) / x)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(z_m * y) - Float64(z_m * t))
	tmp = 0.0
	if (t_1 <= -4e+191)
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / z_m));
	elseif (t_1 <= 2e+220)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(y - t) / x)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (z_m * y) - (z_m * t);
	tmp = 0.0;
	if (t_1 <= -4e+191)
		tmp = (x / (y - t)) * (2.0 / z_m);
	elseif (t_1 <= 2e+220)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (2.0 / ((y - t) / x)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * y), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$1, -4e+191], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+220], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.00000000000000029e191

   1. Initial program 80.8%

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

      1. distribute-rgt-out-- 80.8%

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

   3. Simplified 80.8%

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

      1. *-commutative 80.8%

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

      2. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -4.00000000000000029e191 < (-.f64 (*.f64 y z) (*.f64 t z)) < 2e220

   1. Initial program 98.4%

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

      1. distribute-rgt-out-- 98.4%

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

   3. Simplified 98.4%

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

   if 2e220 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 67.4%

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

      1. distribute-rgt-out-- 77.0%

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

   3. Simplified 77.0%

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

      1. add-sqr-sqrt 45.8%

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

      2. *-commutative 45.8%

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

      3. times-frac 58.2%

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

   6. Applied egg-rr 58.2%

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

      1. associate-*r/ 58.3%

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

      2. associate-*l/ 58.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate-*r/ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-*l/ 99.7%

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

      3. clear-num 99.7%

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

      4. associate-*l/ 99.7%

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

      5. metadata-eval 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - z \cdot t \leq -4 \cdot 10^{+191}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;z \cdot y - z \cdot t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{y - t}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x \cdot 2}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (* x 2.0))))
   (*
    z_s
    (if (<= z_m 1.65e-26)
      (/ (* x 2.0) (* z_m (- y t)))
      (* (/ t_1 (- y t)) (/ t_1 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 = sqrt((x * 2.0));
	double tmp;
	if (z_m <= 1.65e-26) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / 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 = sqrt((x * 2.0d0))
    if (z_m <= 1.65d-26) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (t_1 / (y - t)) * (t_1 / 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 = Math.sqrt((x * 2.0));
	double tmp;
	if (z_m <= 1.65e-26) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / 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 = math.sqrt((x * 2.0))
	tmp = 0
	if z_m <= 1.65e-26:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (t_1 / (y - t)) * (t_1 / 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 = sqrt(Float64(x * 2.0))
	tmp = 0.0
	if (z_m <= 1.65e-26)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(t_1 / Float64(y - t)) * Float64(t_1 / 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 = sqrt((x * 2.0));
	tmp = 0.0;
	if (z_m <= 1.65e-26)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (t_1 / (y - t)) * (t_1 / 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[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 1.65e-26], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6499999999999999e-26

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 93.4%

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

   3. Simplified 93.4%

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

   if 1.6499999999999999e-26 < z

   1. Initial program 81.3%

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

      1. distribute-rgt-out-- 84.0%

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

   3. Simplified 84.0%

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

      1. add-sqr-sqrt 43.7%

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

      2. *-commutative 43.7%

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

      3. times-frac 48.5%

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

   6. Applied egg-rr 48.5%

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

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-55} \lor \neg \left(t \leq 3.4 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

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 -9.5e-55) (not (<= t 3.4e+31)))
    (* (/ 2.0 z_m) (/ x (- t)))
    (* (/ x z_m) (/ 2.0 y)))))

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 <= -9.5e-55) || !(t <= 3.4e+31)) {
		tmp = (2.0 / z_m) * (x / -t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= (-9.5d-55)) .or. (.not. (t <= 3.4d+31))) then
        tmp = (2.0d0 / z_m) * (x / -t)
    else
        tmp = (x / z_m) * (2.0d0 / y)
    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 <= -9.5e-55) || !(t <= 3.4e+31)) {
		tmp = (2.0 / z_m) * (x / -t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= -9.5e-55) or not (t <= 3.4e+31):
		tmp = (2.0 / z_m) * (x / -t)
	else:
		tmp = (x / z_m) * (2.0 / y)
	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 <= -9.5e-55) || !(t <= 3.4e+31))
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(-t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	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 <= -9.5e-55) || ~((t <= 3.4e+31)))
		tmp = (2.0 / z_m) * (x / -t);
	else
		tmp = (x / z_m) * (2.0 / y);
	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, -9.5e-55], N[Not[LessEqual[t, 3.4e+31]], $MachinePrecision]], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000006e-55 or 3.3999999999999998e31 < t

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in y around 0 82.0%

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

      1. associate-*r/ 82.0%

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

      2. neg-mul-1 82.0%

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

   9. Simplified 82.0%

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

   if -9.5000000000000006e-55 < t < 3.3999999999999998e31

   1. Initial program 91.4%

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

      1. distribute-rgt-out-- 91.4%

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

   3. Simplified 91.4%

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

      1. times-frac 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around inf 84.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-55} \lor \neg \left(t \leq 3.4 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-51} \lor \neg \left(t \leq 2.7 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

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 -3.4e-51) (not (<= t 2.7e+23)))
    (/ (* x (/ -2.0 t)) z_m)
    (* (/ x z_m) (/ 2.0 y)))))

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 <= -3.4e-51) || !(t <= 2.7e+23)) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= (-3.4d-51)) .or. (.not. (t <= 2.7d+23))) then
        tmp = (x * ((-2.0d0) / t)) / z_m
    else
        tmp = (x / z_m) * (2.0d0 / y)
    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 <= -3.4e-51) || !(t <= 2.7e+23)) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= -3.4e-51) or not (t <= 2.7e+23):
		tmp = (x * (-2.0 / t)) / z_m
	else:
		tmp = (x / z_m) * (2.0 / y)
	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 <= -3.4e-51) || !(t <= 2.7e+23))
		tmp = Float64(Float64(x * Float64(-2.0 / t)) / z_m);
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	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 <= -3.4e-51) || ~((t <= 2.7e+23)))
		tmp = (x * (-2.0 / t)) / z_m;
	else
		tmp = (x / z_m) * (2.0 / y);
	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, -3.4e-51], N[Not[LessEqual[t, 2.7e+23]], $MachinePrecision]], N[(N[(x * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.40000000000000003e-51 or 2.6999999999999999e23 < t

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 42.5%

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

      2. *-commutative 42.5%

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

      3. times-frac 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. associate-*r/ 45.3%

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

      2. associate-*l/ 45.3%

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

      3. add-sqr-sqrt 92.6%

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

      4. associate-*r/ 92.5%

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

   8. Applied egg-rr 92.5%

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

   9. Taylor expanded in y around 0 82.0%

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

       1. associate-*r/ 82.1%

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

       2. *-commutative 82.1%

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

       3. associate-*r/ 82.0%

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

   11. Simplified 82.0%

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

   if -3.40000000000000003e-51 < t < 2.6999999999999999e23

   1. Initial program 91.4%

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

      1. distribute-rgt-out-- 91.4%

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

   3. Simplified 91.4%

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

      1. times-frac 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around inf 84.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-51} \lor \neg \left(t \leq 2.7 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-51} \lor \neg \left(t \leq 1.02 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

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 -6.5e-51) (not (<= t 1.02e+26)))
    (* x (/ (/ -2.0 t) z_m))
    (* (/ x z_m) (/ 2.0 y)))))

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 <= -6.5e-51) || !(t <= 1.02e+26)) {
		tmp = x * ((-2.0 / t) / z_m);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= (-6.5d-51)) .or. (.not. (t <= 1.02d+26))) then
        tmp = x * (((-2.0d0) / t) / z_m)
    else
        tmp = (x / z_m) * (2.0d0 / y)
    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 <= -6.5e-51) || !(t <= 1.02e+26)) {
		tmp = x * ((-2.0 / t) / z_m);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= -6.5e-51) or not (t <= 1.02e+26):
		tmp = x * ((-2.0 / t) / z_m)
	else:
		tmp = (x / z_m) * (2.0 / y)
	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 <= -6.5e-51) || !(t <= 1.02e+26))
		tmp = Float64(x * Float64(Float64(-2.0 / t) / z_m));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	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 <= -6.5e-51) || ~((t <= 1.02e+26)))
		tmp = x * ((-2.0 / t) / z_m);
	else
		tmp = (x / z_m) * (2.0 / y);
	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, -6.5e-51], N[Not[LessEqual[t, 1.02e+26]], $MachinePrecision]], N[(x * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000003e-51 or 1.0200000000000001e26 < t

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 42.5%

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

      2. *-commutative 42.5%

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

      3. times-frac 43.8%

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

   6. Applied egg-rr 43.8%

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

   7. Taylor expanded in y around 0 78.4%

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

      1. mul-1-neg 78.4%

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

      2. distribute-neg-frac 78.4%

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

      3. unpow2 78.4%

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

      4. rem-square-sqrt 79.0%

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

      5. neg-mul-1 79.0%

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

      6. *-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. associate-*l* 79.0%

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

      9. metadata-eval 79.0%

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

      10. times-frac 75.9%

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

   9. Simplified 75.9%

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

       1. clear-num 75.8%

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

       2. frac-times 76.5%

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

       3. metadata-eval 76.5%

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

   11. Applied egg-rr 76.5%

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

       1. associate-*l/ 78.5%

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

       2. associate-/r/ 78.9%

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

       3. associate-/l/ 80.0%

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

   13. Applied egg-rr 80.0%

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

   if -6.5000000000000003e-51 < t < 1.0200000000000001e26

   1. Initial program 91.4%

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

      1. distribute-rgt-out-- 91.4%

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

   3. Simplified 91.4%

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

      1. times-frac 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around inf 84.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-51} \lor \neg \left(t \leq 1.02 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-51} \lor \neg \left(t \leq 10^{+23}\right):\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

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 -6.5e-51) (not (<= t 1e+23)))
    (* -2.0 (/ x (* z_m t)))
    (* (/ x z_m) (/ 2.0 y)))))

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 <= -6.5e-51) || !(t <= 1e+23)) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= (-6.5d-51)) .or. (.not. (t <= 1d+23))) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else
        tmp = (x / z_m) * (2.0d0 / y)
    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 <= -6.5e-51) || !(t <= 1e+23)) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	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 <= -6.5e-51) or not (t <= 1e+23):
		tmp = -2.0 * (x / (z_m * t))
	else:
		tmp = (x / z_m) * (2.0 / y)
	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 <= -6.5e-51) || !(t <= 1e+23))
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	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 <= -6.5e-51) || ~((t <= 1e+23)))
		tmp = -2.0 * (x / (z_m * t));
	else
		tmp = (x / z_m) * (2.0 / y);
	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, -6.5e-51], N[Not[LessEqual[t, 1e+23]], $MachinePrecision]], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000003e-51 or 9.9999999999999992e22 < t

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 79.0%

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

      1. *-commutative 79.0%

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

   7. Simplified 79.0%

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

   if -6.5000000000000003e-51 < t < 9.9999999999999992e22

   1. Initial program 91.4%

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

      1. distribute-rgt-out-- 91.4%

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

   3. Simplified 91.4%

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

      1. times-frac 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around inf 84.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-51} \lor \neg \left(t \leq 10^{+23}\right):\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-51} \lor \neg \left(t \leq 2.15 \cdot 10^{-32}\right):\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z\_m}\\ \end{array} \end{array} \]

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 (or (<= t -5.5e-51) (not (<= t 2.15e-32)))
    (* -2.0 (/ x (* z_m t)))
    (* 2.0 (/ (/ x y) 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 <= -5.5e-51) || !(t <= 2.15e-32)) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = 2.0 * ((x / y) / 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 <= (-5.5d-51)) .or. (.not. (t <= 2.15d-32))) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else
        tmp = 2.0d0 * ((x / y) / 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 <= -5.5e-51) || !(t <= 2.15e-32)) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = 2.0 * ((x / y) / 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 <= -5.5e-51) or not (t <= 2.15e-32):
		tmp = -2.0 * (x / (z_m * t))
	else:
		tmp = 2.0 * ((x / y) / 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 <= -5.5e-51) || !(t <= 2.15e-32))
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / y) / 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 <= -5.5e-51) || ~((t <= 2.15e-32)))
		tmp = -2.0 * (x / (z_m * t));
	else
		tmp = 2.0 * ((x / y) / 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[Or[LessEqual[t, -5.5e-51], N[Not[LessEqual[t, 2.15e-32]], $MachinePrecision]], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -5.4999999999999997e-51 or 2.14999999999999995e-32 < t

   1. Initial program 86.5%

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

      1. distribute-rgt-out-- 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   if -5.4999999999999997e-51 < t < 2.14999999999999995e-32

   1. Initial program 91.3%

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

      1. distribute-rgt-out-- 91.3%

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

   3. Simplified 91.3%

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

      1. add-sqr-sqrt 50.2%

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

      2. *-commutative 50.2%

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

      3. times-frac 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in y around inf 83.3%

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

      1. *-commutative 83.3%

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

      2. unpow2 83.3%

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

      3. rem-square-sqrt 84.0%

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

      4. associate-*r/ 84.0%

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

      5. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-51} \lor \neg \left(t \leq 2.15 \cdot 10^{-32}\right):\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

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) 5e-42)
    (* (/ x z_m) (/ 2.0 (- y t)))
    (* (/ x (- y t)) (/ 2.0 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) <= 5e-42) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 ((x * 2.0d0) <= 5d-42) then
        tmp = (x / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x / (y - t)) * (2.0d0 / 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 ((x * 2.0) <= 5e-42) {
		tmp = (x / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 (x * 2.0) <= 5e-42:
		tmp = (x / z_m) * (2.0 / (y - t))
	else:
		tmp = (x / (y - t)) * (2.0 / 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) <= 5e-42)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / 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 ((x * 2.0) <= 5e-42)
		tmp = (x / z_m) * (2.0 / (y - t));
	else
		tmp = (x / (y - t)) * (2.0 / 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[(x * 2.0), $MachinePrecision], 5e-42], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5.00000000000000003e-42

   1. Initial program 90.4%

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

      1. distribute-rgt-out-- 92.6%

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

   3. Simplified 92.6%

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

      1. times-frac 92.2%

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

   6. Applied egg-rr 92.2%

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

   if 5.00000000000000003e-42 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.4%

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

      1. distribute-rgt-out-- 85.8%

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

   3. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. times-frac 94.6%

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

   6. Applied egg-rr 94.6%

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

Alternative 9: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \end{array} \end{array} \]

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 (<= z_m 2.9e-36)
    (/ (* x 2.0) (* z_m (- y t)))
    (* 2.0 (/ (/ x z_m) (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.9e-36) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.9d-36) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = 2.0d0 * ((x / z_m) / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.9e-36) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2.9e-36:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = 2.0 * ((x / z_m) / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.9e-36)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.9e-36)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = 2.0 * ((x / z_m) / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2.9e-36], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.90000000000000013e-36

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 93.3%

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

   3. Simplified 93.3%

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

   if 2.90000000000000013e-36 < z

   1. Initial program 81.8%

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

      1. distribute-rgt-out-- 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in x around 0 84.3%

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

      1. associate-/r* 94.9%

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

   7. Simplified 94.9%

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

Alternative 10: 92.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 90.6%

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

3. Simplified 90.6%

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

   1. times-frac 89.6%

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

6. Applied egg-rr 89.6%

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

Alternative 11: 92.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 90.6%

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

3. Simplified 90.6%

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

5. Taylor expanded in x around 0 90.6%

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

   1. associate-/r* 89.6%

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

7. Simplified 89.6%

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

Alternative 12: 52.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 90.6%

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

3. Simplified 90.6%

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

5. Taylor expanded in y around 0 49.5%

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

   1. *-commutative 49.5%

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

7. Simplified 49.5%

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

Developer target: 97.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))