Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.0% → 96.2%

Time: 10.9s

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: 90.0% 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: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \end{array} \end{array} \]

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 7.5e+91)
    (/ (* 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 <= 7.5e+91) {
		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 <= 7.5d+91) 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 <= 7.5e+91) {
		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 <= 7.5e+91:
		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 <= 7.5e+91)
		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 <= 7.5e+91)
		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, 7.5e+91], 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 < 7.50000000000000033e91

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   if 7.50000000000000033e91 < z

   1. Initial program 66.5%

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

      1. distribute-rgt-out-- 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in x around 0 69.1%

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

      1. associate-/r* 97.5%

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

   7. Simplified 97.5%

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

Alternative 2: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-73} \lor \neg \left(t \leq 5.8 \cdot 10^{+44}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

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 -7.5e-73) (not (<= t 5.8e+44)))
    (* -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 <= -7.5e-73) || !(t <= 5.8e+44)) {
		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 <= (-7.5d-73)) .or. (.not. (t <= 5.8d+44))) 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 <= -7.5e-73) || !(t <= 5.8e+44)) {
		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 <= -7.5e-73) or not (t <= 5.8e+44):
		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 <= -7.5e-73) || !(t <= 5.8e+44))
		tmp = Float64(-2.0 * Float64(Float64(x / 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 <= -7.5e-73) || ~((t <= 5.8e+44)))
		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, -7.5e-73], N[Not[LessEqual[t, 5.8e+44]], $MachinePrecision]], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / 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 < -7.5e-73 or 5.8000000000000004e44 < t

   1. Initial program 87.2%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in y around 0 74.2%

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

      1. *-commutative 74.2%

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

   7. Simplified 74.2%

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

      1. associate-/r* 77.5%

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

      2. div-inv 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*r/ 77.5%

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

       2. *-rgt-identity 77.5%

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

   11. Simplified 77.5%

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

   if -7.5e-73 < t < 5.8000000000000004e44

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 78.9%

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

      1. *-commutative 78.9%

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

   7. Simplified 78.9%

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

      1. times-frac 82.5%

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

   9. Applied egg-rr 82.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-73} \lor \neg \left(t \leq 5.8 \cdot 10^{+44}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-72} \lor \neg \left(t \leq 3.6 \cdot 10^{+44}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

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 -4.5e-72) (not (<= t 3.6e+44)))
    (* -2.0 (/ (/ x z_m) t))
    (* (/ 2.0 z_m) (/ x 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 <= -4.5e-72) || !(t <= 3.6e+44)) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (2.0 / z_m) * (x / 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 <= (-4.5d-72)) .or. (.not. (t <= 3.6d+44))) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else
        tmp = (2.0d0 / z_m) * (x / 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 <= -4.5e-72) || !(t <= 3.6e+44)) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (2.0 / z_m) * (x / 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 <= -4.5e-72) or not (t <= 3.6e+44):
		tmp = -2.0 * ((x / z_m) / t)
	else:
		tmp = (2.0 / z_m) * (x / 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 <= -4.5e-72) || !(t <= 3.6e+44))
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / 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 <= -4.5e-72) || ~((t <= 3.6e+44)))
		tmp = -2.0 * ((x / z_m) / t);
	else
		tmp = (2.0 / z_m) * (x / 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, -4.5e-72], N[Not[LessEqual[t, 3.6e+44]], $MachinePrecision]], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -4.5e-72 or 3.6e44 < t

   1. Initial program 87.2%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in y around 0 74.2%

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

      1. *-commutative 74.2%

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

   7. Simplified 74.2%

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

      1. associate-/r* 77.5%

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

      2. div-inv 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*r/ 77.5%

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

       2. *-rgt-identity 77.5%

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

   11. Simplified 77.5%

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

   if -4.5e-72 < t < 3.6e44

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. times-frac 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in y around inf 81.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-72} \lor \neg \left(t \leq 3.6 \cdot 10^{+44}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-73}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{y \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -9.5e-73)
    (* -2.0 (/ (/ x z_m) t))
    (if (<= t 3.8e+44) (/ (/ x z_m) (* y 0.5)) (/ -2.0 (* t (/ z_m x)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -9.5e-73) {
		tmp = -2.0 * ((x / z_m) / t);
	} else if (t <= 3.8e+44) {
		tmp = (x / z_m) / (y * 0.5);
	} else {
		tmp = -2.0 / (t * (z_m / x));
	}
	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-73)) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else if (t <= 3.8d+44) then
        tmp = (x / z_m) / (y * 0.5d0)
    else
        tmp = (-2.0d0) / (t * (z_m / x))
    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-73) {
		tmp = -2.0 * ((x / z_m) / t);
	} else if (t <= 3.8e+44) {
		tmp = (x / z_m) / (y * 0.5);
	} else {
		tmp = -2.0 / (t * (z_m / x));
	}
	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-73:
		tmp = -2.0 * ((x / z_m) / t)
	elif t <= 3.8e+44:
		tmp = (x / z_m) / (y * 0.5)
	else:
		tmp = -2.0 / (t * (z_m / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -9.5e-73)
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	elseif (t <= 3.8e+44)
		tmp = Float64(Float64(x / z_m) / Float64(y * 0.5));
	else
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	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-73)
		tmp = -2.0 * ((x / z_m) / t);
	elseif (t <= 3.8e+44)
		tmp = (x / z_m) / (y * 0.5);
	else
		tmp = -2.0 / (t * (z_m / x));
	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, -9.5e-73], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e+44], N[(N[(x / z$95$m), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -9.50000000000000005e-73

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in y around 0 71.7%

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

      1. *-commutative 71.7%

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

   7. Simplified 71.7%

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

      1. associate-/r* 76.3%

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

      2. div-inv 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.3%

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

       2. *-rgt-identity 76.3%

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

   11. Simplified 76.3%

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

   if -9.50000000000000005e-73 < t < 3.8000000000000002e44

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 78.9%

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

      1. *-commutative 78.9%

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

   7. Simplified 78.9%

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

      1. times-frac 82.5%

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

   9. Applied egg-rr 82.5%

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

       1. clear-num 82.5%

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

       2. un-div-inv 82.6%

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

       3. div-inv 82.6%

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

       4. metadata-eval 82.6%

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

   11. Applied egg-rr 82.6%

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

   if 3.8000000000000002e44 < t

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. *-commutative 77.3%

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

   7. Simplified 77.3%

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

      1. associate-/r* 79.0%

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

      2. div-inv 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. associate-*r/ 79.0%

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

       2. *-rgt-identity 79.0%

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

   11. Simplified 79.0%

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

       1. clear-num 79.1%

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

       2. un-div-inv 79.1%

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

       3. div-inv 79.1%

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

       4. clear-num 80.3%

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

   13. Applied egg-rr 80.3%

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

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-72}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -2e-72)
    (* -2.0 (/ (/ x z_m) t))
    (if (<= t 4.5e+44) (* (/ x z_m) (/ 2.0 y)) (/ -2.0 (* t (/ z_m x)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -2e-72) {
		tmp = -2.0 * ((x / z_m) / t);
	} else if (t <= 4.5e+44) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 / (t * (z_m / x));
	}
	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 <= (-2d-72)) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else if (t <= 4.5d+44) then
        tmp = (x / z_m) * (2.0d0 / y)
    else
        tmp = (-2.0d0) / (t * (z_m / x))
    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 <= -2e-72) {
		tmp = -2.0 * ((x / z_m) / t);
	} else if (t <= 4.5e+44) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 / (t * (z_m / x));
	}
	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 <= -2e-72:
		tmp = -2.0 * ((x / z_m) / t)
	elif t <= 4.5e+44:
		tmp = (x / z_m) * (2.0 / y)
	else:
		tmp = -2.0 / (t * (z_m / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -2e-72)
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	elseif (t <= 4.5e+44)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	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 <= -2e-72)
		tmp = -2.0 * ((x / z_m) / t);
	elseif (t <= 4.5e+44)
		tmp = (x / z_m) * (2.0 / y);
	else
		tmp = -2.0 / (t * (z_m / x));
	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, -2e-72], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e+44], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e-72

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in y around 0 71.7%

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

      1. *-commutative 71.7%

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

   7. Simplified 71.7%

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

      1. associate-/r* 76.3%

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

      2. div-inv 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.3%

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

       2. *-rgt-identity 76.3%

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

   11. Simplified 76.3%

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

   if -1.9999999999999999e-72 < t < 4.5e44

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 78.9%

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

      1. *-commutative 78.9%

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

   7. Simplified 78.9%

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

      1. times-frac 82.5%

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

   9. Applied egg-rr 82.5%

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

   if 4.5e44 < t

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. *-commutative 77.3%

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

   7. Simplified 77.3%

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

      1. associate-/r* 79.0%

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

      2. div-inv 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. associate-*r/ 79.0%

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

       2. *-rgt-identity 79.0%

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

   11. Simplified 79.0%

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

       1. clear-num 79.1%

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

       2. un-div-inv 79.1%

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

       3. div-inv 79.1%

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

       4. clear-num 80.3%

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

   13. Applied egg-rr 80.3%

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

Alternative 6: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y - t}\\ \end{array} \end{array} \]

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 (<= (* x 2.0) 1e-85)
    (* 2.0 (/ (/ x z_m) (- y t)))
    (* (/ 2.0 z_m) (/ x (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-85) {
		tmp = 2.0 * ((x / z_m) / (y - t));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * 2.0d0) <= 1d-85) then
        tmp = 2.0d0 * ((x / z_m) / (y - t))
    else
        tmp = (2.0d0 / z_m) * (x / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1e-85) {
		tmp = 2.0 * ((x / z_m) / (y - t));
	} else {
		tmp = (2.0 / z_m) * (x / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (x * 2.0) <= 1e-85:
		tmp = 2.0 * ((x / z_m) / (y - t))
	else:
		tmp = (2.0 / z_m) * (x / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 1e-85)
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((x * 2.0) <= 1e-85)
		tmp = 2.0 * ((x / z_m) / (y - t));
	else
		tmp = (2.0 / z_m) * (x / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 1e-85], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.9999999999999998e-86

   1. Initial program 87.3%

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

      1. distribute-rgt-out-- 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around 0 89.6%

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

      1. associate-/r* 96.6%

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

   7. Simplified 96.6%

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

   if 9.9999999999999998e-86 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 91.1%

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

   3. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. times-frac 96.1%

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

   6. Applied egg-rr 96.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \end{array} \end{array} \]

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 7e+91)
    (* 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 <= 7e+91) {
		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 <= 7d+91) 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 <= 7e+91) {
		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 <= 7e+91:
		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 <= 7e+91)
		tmp = Float64(x * Float64(Float64(2.0 / 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 <= 7e+91)
		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, 7e+91], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / 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 < 7.00000000000000001e91

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

      1. add-sqr-sqrt 50.8%

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

      2. *-commutative 50.8%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 49.7%

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

      1. frac-times 50.8%

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

      2. add-sqr-sqrt 94.4%

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

      3. frac-times 89.4%

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

      4. clear-num 89.0%

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

      5. frac-times 88.9%

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

      6. metadata-eval 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. associate-/l/ 89.1%

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

      2. associate-/r/ 94.7%

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

   10. Simplified 94.7%

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

   if 7.00000000000000001e91 < z

   1. Initial program 66.5%

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

      1. distribute-rgt-out-- 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in x around 0 69.1%

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

      1. associate-/r* 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7 \cdot 10^{+97}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

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 7e+97) (* -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 <= 7e+97) {
		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 <= 7d+97) 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 <= 7e+97) {
		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 <= 7e+97:
		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 <= 7e+97)
		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 <= 7e+97)
		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, 7e+97], 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 < 7.0000000000000001e97

   1. Initial program 92.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. Taylor expanded in y around 0 55.8%

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

      1. *-commutative 55.8%

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

   7. Simplified 55.8%

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

   if 7.0000000000000001e97 < z

   1. Initial program 67.3%

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

      1. distribute-rgt-out-- 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in y around 0 46.9%

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

      1. *-commutative 46.9%

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

   7. Simplified 46.9%

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

      1. associate-/r* 69.7%

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

      2. div-inv 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-*r/ 69.7%

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

       2. *-rgt-identity 69.7%

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

   11. Simplified 69.7%

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

Alternative 9: 91.8% accurate, 1.2× speedup

Math

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

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 88.5%

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

   1. distribute-rgt-out-- 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in x around 0 90.1%

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

   1. associate-/r* 93.6%

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

7. Simplified 93.6%

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

Alternative 10: 52.9% 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 88.5%

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

   1. distribute-rgt-out-- 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in y around 0 54.4%

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

   1. *-commutative 54.4%

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

7. Simplified 54.4%

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

Developer target: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024085 
(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))))