Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 96.4%

Time: 10.5s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{+59}:\\ \;\;\;\;\frac{-2 \cdot \frac{x\_m}{z}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - t}}{z \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 1e+59)
    (/ (* -2.0 (/ x_m z)) (- t y))
    (/ (/ x_m (- y t)) (* z 0.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+59) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) / (z * 0.5);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 1d+59) then
        tmp = ((-2.0d0) * (x_m / z)) / (t - y)
    else
        tmp = (x_m / (y - t)) / (z * 0.5d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+59) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) / (z * 0.5);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 1e+59:
		tmp = (-2.0 * (x_m / z)) / (t - y)
	else:
		tmp = (x_m / (y - t)) / (z * 0.5)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 1e+59)
		tmp = Float64(Float64(-2.0 * Float64(x_m / z)) / Float64(t - y));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) / Float64(z * 0.5));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 1e+59)
		tmp = (-2.0 * (x_m / z)) / (t - y);
	else
		tmp = (x_m / (y - t)) / (z * 0.5);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 1e+59], N[(N[(-2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.99999999999999972e58

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 93.2%

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

      1. associate-*r/ 93.2%

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

      2. metadata-eval 93.2%

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

      3. distribute-lft-neg-in 93.2%

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

      4. *-commutative 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. associate-/r* 96.2%

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

      7. *-commutative 96.2%

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

      8. associate-*r/ 96.2%

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

      9. distribute-neg-frac2 96.2%

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

      10. neg-sub0 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{0 - \left(y - t\right)}} \]

      11. sub-neg 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      13. associate--r+ 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      14. neg-sub0 96.2%

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

      15. remove-double-neg 96.2%

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

   7. Simplified 96.2%

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

   if 9.99999999999999972e58 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 81.1%

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

      1. distribute-rgt-out-- 81.2%

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

   3. Simplified 81.2%

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

      1. *-commutative 81.2%

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

      2. times-frac 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-2neg 99.7%

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

      4. frac-times 81.2%

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

      5. *-un-lft-identity 81.2%

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

      6. div-inv 81.2%

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

      7. metadata-eval 81.2%

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

      8. sub-neg 81.2%

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

      9. distribute-neg-in 81.2%

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

      10. add-sqr-sqrt 36.7%

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

      11. sqrt-unprod 58.5%

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

      12. sqr-neg 58.5%

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

      13. sqrt-unprod 26.1%

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

      14. add-sqr-sqrt 53.8%

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

      15. add-sqr-sqrt 27.6%

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

      16. sqrt-unprod 65.4%

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

      17. sqr-neg 65.4%

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

      18. sqrt-unprod 44.5%

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

      19. add-sqr-sqrt 81.2%

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

   8. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\frac{-x}{\left(z \cdot 0.5\right) \cdot \left(\left(-y\right) + t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 81.2%

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

      2. sub-neg 81.2%

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

      3. *-commutative 81.2%

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

      4. sub-neg 81.2%

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

      5. +-commutative 81.2%

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

      6. associate-/r* 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. distribute-neg-frac2 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. remove-double-neg 99.8%

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

      11. sub-neg 99.8%

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

   10. Simplified 99.8%

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

Alternative 2: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-81}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{x\_m}{z}}{-y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -2.8e+33)
    (* (/ 2.0 z) (/ x_m y))
    (if (<= y 4.4e-81)
      (* -2.0 (/ (/ x_m z) t))
      (/ (* -2.0 (/ x_m z)) (- y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e+33) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 4.4e-81) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (-2.0 * (x_m / z)) / -y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.8d+33)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= 4.4d-81) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = ((-2.0d0) * (x_m / z)) / -y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e+33) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 4.4e-81) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (-2.0 * (x_m / z)) / -y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -2.8e+33:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= 4.4e-81:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = (-2.0 * (x_m / z)) / -y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -2.8e+33)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= 4.4e-81)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(Float64(-2.0 * Float64(x_m / z)) / Float64(-y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -2.8e+33)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= 4.4e-81)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = (-2.0 * (x_m / z)) / -y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -2.8e+33], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e-81], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e33

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. times-frac 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in y around inf 87.1%

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

   if -2.8000000000000001e33 < y < 4.3999999999999998e-81

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/r* 82.1%

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

   7. Simplified 82.1%

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

   if 4.3999999999999998e-81 < y

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 89.7%

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

      1. associate-*r/ 89.8%

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

      2. metadata-eval 89.8%

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

      3. distribute-lft-neg-in 89.8%

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

      4. *-commutative 89.8%

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

      5. distribute-neg-frac 89.8%

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

      6. associate-/r* 92.2%

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

      7. *-commutative 92.2%

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

      8. associate-*r/ 92.2%

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

      9. distribute-neg-frac2 92.2%

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

      10. neg-sub0 92.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{0 - \left(y - t\right)}} \]

      11. sub-neg 92.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      12. +-commutative 92.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      13. associate--r+ 92.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      14. neg-sub0 92.2%

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

      15. remove-double-neg 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in t around 0 75.1%

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

      1. neg-mul-1 75.1%

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

   10. Simplified 75.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-81}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{x}{z}}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+34} \lor \neg \left(y \leq 7.2 \cdot 10^{-84}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -5e+34) (not (<= y 7.2e-84)))
    (* 2.0 (/ (/ x_m z) y))
    (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -5e+34) || !(y <= 7.2e-84)) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-5d+34)) .or. (.not. (y <= 7.2d-84))) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -5e+34) || !(y <= 7.2e-84)) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -5e+34) or not (y <= 7.2e-84):
		tmp = 2.0 * ((x_m / z) / y)
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -5e+34) || !(y <= 7.2e-84))
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -5e+34) || ~((y <= 7.2e-84)))
		tmp = 2.0 * ((x_m / z) / y);
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -5e+34], N[Not[LessEqual[y, 7.2e-84]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999998e34 or 7.20000000000000007e-84 < y

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. times-frac 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. *-commutative 92.3%

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

      2. clear-num 92.3%

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

      3. frac-2neg 92.3%

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

      4. frac-times 89.7%

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

      5. *-un-lft-identity 89.7%

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

      6. div-inv 89.7%

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

      7. metadata-eval 89.7%

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

      8. sub-neg 89.7%

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

      9. distribute-neg-in 89.7%

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

      10. add-sqr-sqrt 43.8%

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

      11. sqrt-unprod 77.1%

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

      12. sqr-neg 77.1%

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

      13. sqrt-unprod 41.4%

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

      14. add-sqr-sqrt 77.4%

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

      15. add-sqr-sqrt 36.0%

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

      16. sqrt-unprod 76.1%

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

      17. sqr-neg 76.1%

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

      18. sqrt-unprod 45.9%

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

      19. add-sqr-sqrt 89.7%

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

   8. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{-x}{\left(z \cdot 0.5\right) \cdot \left(\left(-y\right) + t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 89.7%

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

      2. sub-neg 89.7%

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

      3. *-commutative 89.7%

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

      4. sub-neg 89.7%

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

      5. +-commutative 89.7%

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

      6. associate-/r* 92.3%

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

      7. distribute-neg-frac 92.3%

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

      8. distribute-neg-frac2 92.3%

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

      9. distribute-neg-in 92.3%

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

      10. remove-double-neg 92.3%

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

      11. sub-neg 92.3%

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

   10. Simplified 92.3%

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

   11. Taylor expanded in y around inf 75.8%

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

       1. *-commutative 75.8%

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

       2. associate-/r* 76.7%

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

   13. Simplified 76.7%

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

   if -4.9999999999999998e34 < y < 7.20000000000000007e-84

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/r* 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+34} \lor \neg \left(y \leq 7.2 \cdot 10^{-84}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.3e+33)
    (* (/ 2.0 z) (/ x_m y))
    (if (<= y 1.4e-76) (* -2.0 (/ (/ x_m z) t)) (* 2.0 (/ (/ x_m z) y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.3e+33) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 1.4e-76) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = 2.0 * ((x_m / z) / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.3d+33)) then
        tmp = (2.0d0 / z) * (x_m / y)
    else if (y <= 1.4d-76) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = 2.0d0 * ((x_m / z) / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.3e+33) {
		tmp = (2.0 / z) * (x_m / y);
	} else if (y <= 1.4e-76) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = 2.0 * ((x_m / z) / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -3.3e+33:
		tmp = (2.0 / z) * (x_m / y)
	elif y <= 1.4e-76:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = 2.0 * ((x_m / z) / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -3.3e+33)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	elseif (y <= 1.4e-76)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -3.3e+33)
		tmp = (2.0 / z) * (x_m / y);
	elseif (y <= 1.4e-76)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = 2.0 * ((x_m / z) / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -3.3e+33], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-76], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.29999999999999976e33

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. times-frac 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in y around inf 87.1%

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

   if -3.29999999999999976e33 < y < 1.40000000000000005e-76

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/r* 82.1%

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

   7. Simplified 82.1%

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

   if 1.40000000000000005e-76 < y

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. *-commutative 91.7%

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

      2. clear-num 91.7%

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

      3. frac-2neg 91.7%

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

      4. frac-times 89.8%

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

      5. *-un-lft-identity 89.8%

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

      6. div-inv 89.8%

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

      7. metadata-eval 89.8%

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

      8. sub-neg 89.8%

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

      9. distribute-neg-in 89.8%

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

      10. add-sqr-sqrt 49.7%

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

      11. sqrt-unprod 75.8%

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

      12. sqr-neg 75.8%

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

      13. sqrt-unprod 35.1%

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

      14. add-sqr-sqrt 73.8%

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

      15. add-sqr-sqrt 38.7%

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

      16. sqrt-unprod 72.2%

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

      17. sqr-neg 72.2%

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

      18. sqrt-unprod 40.0%

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

      19. add-sqr-sqrt 89.8%

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

   8. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\frac{-x}{\left(z \cdot 0.5\right) \cdot \left(\left(-y\right) + t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 89.8%

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

      2. sub-neg 89.8%

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

      3. *-commutative 89.8%

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

      4. sub-neg 89.8%

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

      5. +-commutative 89.8%

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

      6. associate-/r* 91.8%

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

      7. distribute-neg-frac 91.8%

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

      8. distribute-neg-frac2 91.8%

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

      9. distribute-neg-in 91.8%

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

      10. remove-double-neg 91.8%

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

      11. sub-neg 91.8%

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

   10. Simplified 91.8%

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

   11. Taylor expanded in y around inf 72.2%

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

       1. *-commutative 72.2%

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

       2. associate-/r* 75.1%

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

   13. Simplified 75.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 1e+65)
    (/ (* -2.0 (/ x_m z)) (- t y))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+65) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 1d+65) then
        tmp = ((-2.0d0) * (x_m / z)) / (t - y)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+65) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 1e+65:
		tmp = (-2.0 * (x_m / z)) / (t - y)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 1e+65)
		tmp = Float64(Float64(-2.0 * Float64(x_m / z)) / Float64(t - y));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 1e+65)
		tmp = (-2.0 * (x_m / z)) / (t - y);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 1e+65], N[(N[(-2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.9999999999999999e64

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around 0 93.3%

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

      1. associate-*r/ 93.3%

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

      2. metadata-eval 93.3%

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

      3. distribute-lft-neg-in 93.3%

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

      4. *-commutative 93.3%

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

      5. distribute-neg-frac 93.3%

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

      6. associate-/r* 96.2%

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

      7. *-commutative 96.2%

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

      8. associate-*r/ 96.2%

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

      9. distribute-neg-frac2 96.2%

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

      10. neg-sub0 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{0 - \left(y - t\right)}} \]

      11. sub-neg 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      12. +-commutative 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      13. associate--r+ 96.2%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      14. neg-sub0 96.2%

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

      15. remove-double-neg 96.2%

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

   7. Simplified 96.2%

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

   if 9.9999999999999999e64 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 80.2%

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

      1. distribute-rgt-out-- 80.3%

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

   3. Simplified 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 99.7%

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

   6. Applied egg-rr 99.7%

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

Alternative 6: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-84}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 1.55e-84)
    (* x_m (/ 2.0 (* z (- y t))))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-84) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.55d-84) then
        tmp = x_m * (2.0d0 / (z * (y - t)))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-84) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 1.55e-84:
		tmp = x_m * (2.0 / (z * (y - t)))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.55e-84)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * Float64(y - t))));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1.55e-84)
		tmp = x_m * (2.0 / (z * (y - t)));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 1.55e-84], N[(x$95$m * N[(2.0 / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.55000000000000001e-84

   1. Initial program 90.3%

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

      1. distribute-rgt-out-- 90.9%

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

   3. Simplified 90.9%

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

      1. distribute-rgt-out-- 90.3%

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

      2. associate-/l* 89.9%

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

      3. *-commutative 89.9%

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

      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%

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

   if 1.55000000000000001e-84 < z

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

      1. *-commutative 92.0%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. distribute-rgt-out-- 91.2%

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

3. Simplified 91.2%

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

   1. distribute-rgt-out-- 90.4%

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

   2. associate-/l* 90.1%

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

   3. *-commutative 90.1%

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

   4. distribute-rgt-out-- 90.9%

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

6. Applied egg-rr 90.9%

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

7. Final simplification 90.9%

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

Alternative 8: 55.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. distribute-rgt-out-- 91.2%

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

3. Simplified 91.2%

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

5. Taylor expanded in y around 0 54.0%

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

   1. *-commutative 54.0%

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

   2. associate-/r* 57.2%

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

7. Simplified 57.2%

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

Alternative 9: 53.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. distribute-rgt-out-- 91.2%

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

3. Simplified 91.2%

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

5. Taylor expanded in y around 0 54.0%

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

   1. *-commutative 54.0%

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

7. Simplified 54.0%

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

Developer Target 1: 96.8% 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 2024180 
(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))))