Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.9% → 97.2%

Time: 13.7s

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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 20:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

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 20.0)
    (/ (* x 2.0) (* z_m (- y t)))
    (* (/ 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) {
	double tmp;
	if (z_m <= 20.0) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (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 <= 20.0d0) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x / z_m) * (2.0d0 / (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 <= 20.0) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (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 <= 20.0:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (x / z_m) * (2.0 / (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 <= 20.0)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / 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 <= 20.0)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (x / z_m) * (2.0 / (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, 20.0], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 20

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

   if 20 < z

   1. Initial program 80.2%

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

      1. distribute-rgt-out-- 82.8%

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

   3. Simplified 82.8%

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

      1. times-frac 98.0%

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

   6. Applied egg-rr 98.0%

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

Alternative 2: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{-2}{z\_m \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y + t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 (<= t -7.6e+42)
    (/ -2.0 (* z_m (/ t x)))
    (if (<= t 1.12e-33)
      (* x (/ (/ 2.0 (+ y t)) z_m))
      (* -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) {
	double tmp;
	if (t <= -7.6e+42) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 1.12e-33) {
		tmp = x * ((2.0 / (y + t)) / z_m);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 (t <= (-7.6d+42)) then
        tmp = (-2.0d0) / (z_m * (t / x))
    else if (t <= 1.12d-33) then
        tmp = x * ((2.0d0 / (y + t)) / z_m)
    else
        tmp = (-2.0d0) * ((x / z_m) / 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 (t <= -7.6e+42) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 1.12e-33) {
		tmp = x * ((2.0 / (y + t)) / z_m);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 t <= -7.6e+42:
		tmp = -2.0 / (z_m * (t / x))
	elif t <= 1.12e-33:
		tmp = x * ((2.0 / (y + t)) / z_m)
	else:
		tmp = -2.0 * ((x / z_m) / 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 (t <= -7.6e+42)
		tmp = Float64(-2.0 / Float64(z_m * Float64(t / x)));
	elseif (t <= 1.12e-33)
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y + t)) / z_m));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / 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 (t <= -7.6e+42)
		tmp = -2.0 / (z_m * (t / x));
	elseif (t <= 1.12e-33)
		tmp = x * ((2.0 / (y + t)) / z_m);
	else
		tmp = -2.0 * ((x / z_m) / 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[t, -7.6e+42], N[(-2.0 / N[(z$95$m * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.12e-33], N[(x * N[(N[(2.0 / N[(y + t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.5999999999999997e42

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

      1. times-frac 87.8%

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

   6. Applied egg-rr 87.8%

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

   7. Taylor expanded in y around 0 76.9%

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

      1. clear-num 77.0%

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

      2. frac-times 75.9%

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

      3. metadata-eval 75.9%

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

   9. Applied egg-rr 75.9%

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

   10. Taylor expanded in z around 0 72.7%

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

       1. *-commutative 72.7%

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

       2. associate-*r/ 79.1%

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

   12. Simplified 79.1%

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

   if -7.5999999999999997e42 < t < 1.11999999999999999e-33

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. times-frac 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. clear-num 91.3%

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

      2. frac-times 91.9%

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

      3. metadata-eval 91.9%

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

      4. sub-neg 91.9%

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

      5. add-sqr-sqrt 60.4%

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

      6. sqrt-unprod 82.0%

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

      7. sqr-neg 82.0%

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

      8. sqrt-unprod 25.8%

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

      9. add-sqr-sqrt 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. associate-*l/ 74.6%

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

      2. associate-/r/ 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-/l/ 75.7%

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

   10. Simplified 75.7%

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

   if 1.11999999999999999e-33 < t

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+30} \lor \neg \left(t \leq 1.36 \cdot 10^{-33}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y}\\ \end{array} \end{array} \]

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 -1.5e+30) (not (<= t 1.36e-33)))
    (* -2.0 (/ (/ x z_m) t))
    (* 2.0 (/ (/ x z_m) 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 <= -1.5e+30) || !(t <= 1.36e-33)) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = 2.0 * ((x / z_m) / 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 <= (-1.5d+30)) .or. (.not. (t <= 1.36d-33))) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else
        tmp = 2.0d0 * ((x / z_m) / 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 <= -1.5e+30) || !(t <= 1.36e-33)) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = 2.0 * ((x / z_m) / 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 <= -1.5e+30) or not (t <= 1.36e-33):
		tmp = -2.0 * ((x / z_m) / t)
	else:
		tmp = 2.0 * ((x / z_m) / 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 <= -1.5e+30) || !(t <= 1.36e-33))
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	else
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / 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 <= -1.5e+30) || ~((t <= 1.36e-33)))
		tmp = -2.0 * ((x / z_m) / t);
	else
		tmp = 2.0 * ((x / z_m) / 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, -1.5e+30], N[Not[LessEqual[t, 1.36e-33]], $MachinePrecision]], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -1.49999999999999989e30 or 1.36e-33 < t

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. associate-/r* 78.5%

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

   7. Simplified 78.5%

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

   if -1.49999999999999989e30 < t < 1.36e-33

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

      1. times-frac 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in y around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-/r* 74.6%

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

   9. Simplified 74.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+30} \lor \neg \left(t \leq 1.36 \cdot 10^{-33}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{-2}{z\_m \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 (<= t -7.6e+42)
    (/ -2.0 (* z_m (/ t x)))
    (if (<= t 3.3e-34) (* x (/ (/ 2.0 z_m) y)) (* -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) {
	double tmp;
	if (t <= -7.6e+42) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 3.3e-34) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 (t <= (-7.6d+42)) then
        tmp = (-2.0d0) / (z_m * (t / x))
    else if (t <= 3.3d-34) then
        tmp = x * ((2.0d0 / z_m) / y)
    else
        tmp = (-2.0d0) * ((x / z_m) / 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 (t <= -7.6e+42) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 3.3e-34) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 t <= -7.6e+42:
		tmp = -2.0 / (z_m * (t / x))
	elif t <= 3.3e-34:
		tmp = x * ((2.0 / z_m) / y)
	else:
		tmp = -2.0 * ((x / z_m) / 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 (t <= -7.6e+42)
		tmp = Float64(-2.0 / Float64(z_m * Float64(t / x)));
	elseif (t <= 3.3e-34)
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / 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 (t <= -7.6e+42)
		tmp = -2.0 / (z_m * (t / x));
	elseif (t <= 3.3e-34)
		tmp = x * ((2.0 / z_m) / y);
	else
		tmp = -2.0 * ((x / z_m) / 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[t, -7.6e+42], N[(-2.0 / N[(z$95$m * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-34], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.5999999999999997e42

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

      1. times-frac 87.8%

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

   6. Applied egg-rr 87.8%

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

   7. Taylor expanded in y around 0 76.9%

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

      1. clear-num 77.0%

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

      2. frac-times 75.9%

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

      3. metadata-eval 75.9%

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

   9. Applied egg-rr 75.9%

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

   10. Taylor expanded in z around 0 72.7%

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

       1. *-commutative 72.7%

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

       2. associate-*r/ 79.1%

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

   12. Simplified 79.1%

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

   if -7.5999999999999997e42 < t < 3.29999999999999983e-34

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. distribute-rgt-out-- 93.1%

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

      2. associate-/l* 93.1%

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

      3. *-commutative 93.1%

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

      4. distribute-rgt-out-- 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in y around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/r* 74.4%

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

   9. Simplified 74.4%

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

   if 3.29999999999999983e-34 < t

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 (<= t -1e+43)
    (* (/ x z_m) (/ -2.0 t))
    (if (<= t 2e-34) (* x (/ (/ 2.0 z_m) y)) (* -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) {
	double tmp;
	if (t <= -1e+43) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if (t <= 2e-34) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 (t <= (-1d+43)) then
        tmp = (x / z_m) * ((-2.0d0) / t)
    else if (t <= 2d-34) then
        tmp = x * ((2.0d0 / z_m) / y)
    else
        tmp = (-2.0d0) * ((x / z_m) / 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 (t <= -1e+43) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if (t <= 2e-34) {
		tmp = x * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 t <= -1e+43:
		tmp = (x / z_m) * (-2.0 / t)
	elif t <= 2e-34:
		tmp = x * ((2.0 / z_m) / y)
	else:
		tmp = -2.0 * ((x / z_m) / 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 (t <= -1e+43)
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	elseif (t <= 2e-34)
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / 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 (t <= -1e+43)
		tmp = (x / z_m) * (-2.0 / t);
	elseif (t <= 2e-34)
		tmp = x * ((2.0 / z_m) / y);
	else
		tmp = -2.0 * ((x / z_m) / 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[t, -1e+43], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-34], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000001e43

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

      1. times-frac 87.8%

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

   6. Applied egg-rr 87.8%

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

   7. Taylor expanded in y around 0 76.9%

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

   if -1.00000000000000001e43 < t < 1.99999999999999986e-34

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. distribute-rgt-out-- 93.1%

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

      2. associate-/l* 93.1%

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

      3. *-commutative 93.1%

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

      4. distribute-rgt-out-- 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in y around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/r* 74.4%

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

   9. Simplified 74.4%

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

   if 1.99999999999999986e-34 < t

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 (<= t -5.2e+29)
    (* (/ x z_m) (/ -2.0 t))
    (if (<= t 2.2e-33) (* 2.0 (/ (/ x z_m) y)) (* -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) {
	double tmp;
	if (t <= -5.2e+29) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if (t <= 2.2e-33) {
		tmp = 2.0 * ((x / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 (t <= (-5.2d+29)) then
        tmp = (x / z_m) * ((-2.0d0) / t)
    else if (t <= 2.2d-33) then
        tmp = 2.0d0 * ((x / z_m) / y)
    else
        tmp = (-2.0d0) * ((x / z_m) / 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 (t <= -5.2e+29) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if (t <= 2.2e-33) {
		tmp = 2.0 * ((x / z_m) / y);
	} else {
		tmp = -2.0 * ((x / z_m) / 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 t <= -5.2e+29:
		tmp = (x / z_m) * (-2.0 / t)
	elif t <= 2.2e-33:
		tmp = 2.0 * ((x / z_m) / y)
	else:
		tmp = -2.0 * ((x / z_m) / 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 (t <= -5.2e+29)
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	elseif (t <= 2.2e-33)
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / 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 (t <= -5.2e+29)
		tmp = (x / z_m) * (-2.0 / t);
	elseif (t <= 2.2e-33)
		tmp = 2.0 * ((x / z_m) / y);
	else
		tmp = -2.0 * ((x / z_m) / 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[t, -5.2e+29], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e-33], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.2e29

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 88.6%

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

   3. Simplified 88.6%

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

      1. times-frac 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in y around 0 75.5%

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

   if -5.2e29 < t < 2.20000000000000005e-33

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

      1. times-frac 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in y around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-/r* 74.6%

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

   9. Simplified 74.6%

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

   if 2.20000000000000005e-33 < t

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

Alternative 7: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1000000000000:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

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 1000000000000.0)
    (* x (/ 2.0 (* z_m (- y t))))
    (* (/ 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) {
	double tmp;
	if (z_m <= 1000000000000.0) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / z_m) * (2.0 / (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 <= 1000000000000.0d0) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x / z_m) * (2.0d0 / (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 <= 1000000000000.0) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / z_m) * (2.0 / (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 <= 1000000000000.0:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x / z_m) * (2.0 / (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 <= 1000000000000.0)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / 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 <= 1000000000000.0)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (x / z_m) * (2.0 / (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, 1000000000000.0], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e12

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. distribute-rgt-out-- 91.8%

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

      2. associate-/l* 91.8%

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

      3. *-commutative 91.8%

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

      4. distribute-rgt-out-- 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 1e12 < z

   1. Initial program 79.4%

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

      1. distribute-rgt-out-- 82.1%

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

   3. Simplified 82.1%

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

      1. times-frac 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1000000000000:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.2 \cdot 10^{-39}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 8.2e-39) (* -2.0 (/ x (* z_m t))) (* -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) {
	double tmp;
	if (z_m <= 8.2e-39) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / 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 <= 8.2d-39) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else
        tmp = (-2.0d0) * ((x / z_m) / 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 <= 8.2e-39) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / 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 <= 8.2e-39:
		tmp = -2.0 * (x / (z_m * t))
	else:
		tmp = -2.0 * ((x / z_m) / 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 <= 8.2e-39)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / 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 <= 8.2e-39)
		tmp = -2.0 * (x / (z_m * t));
	else
		tmp = -2.0 * ((x / z_m) / 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, 8.2e-39], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.2e-39

   1. Initial program 91.2%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in y around 0 56.9%

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

      1. *-commutative 56.9%

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

   7. Simplified 56.9%

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

   if 8.2e-39 < z

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in y around 0 47.8%

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

      1. *-commutative 47.8%

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

      2. associate-/r* 60.6%

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

   7. Simplified 60.6%

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

Alternative 9: 91.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. distribute-rgt-out-- 88.2%

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

   2. associate-/l* 88.2%

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

   3. *-commutative 88.2%

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

   4. distribute-rgt-out-- 90.5%

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

6. Applied egg-rr 90.5%

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

7. Final simplification 90.5%

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

Alternative 10: 53.6% 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 88.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 53.8%

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

   1. *-commutative 53.8%

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

7. Simplified 53.8%

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

Developer Target 1: 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 2024181 
(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))))