Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.0% → 96.5%

Time: 10.2s

Alternatives: 8

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: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{y - t}}{z}\\ \end{array} \end{array} \]

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) 2e-82)
    (/ (/ (* x_m 2.0) z) (- y t))
    (/ (/ (* x_m 2.0) (- y t)) 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) <= 2e-82) {
		tmp = ((x_m * 2.0) / z) / (y - t);
	} else {
		tmp = ((x_m * 2.0) / (y - t)) / 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) <= 2d-82) then
        tmp = ((x_m * 2.0d0) / z) / (y - t)
    else
        tmp = ((x_m * 2.0d0) / (y - t)) / 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) <= 2e-82) {
		tmp = ((x_m * 2.0) / z) / (y - t);
	} else {
		tmp = ((x_m * 2.0) / (y - t)) / 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) <= 2e-82:
		tmp = ((x_m * 2.0) / z) / (y - t)
	else:
		tmp = ((x_m * 2.0) / (y - t)) / 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) <= 2e-82)
		tmp = Float64(Float64(Float64(x_m * 2.0) / z) / Float64(y - t));
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / Float64(y - t)) / 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) <= 2e-82)
		tmp = ((x_m * 2.0) / z) / (y - t);
	else
		tmp = ((x_m * 2.0) / (y - t)) / 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], 2e-82], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2e-82

   1. Initial program 89.5%

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

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

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. --lowering--.f64 96.3

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

   4. Applied egg-rr 96.3%

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

   if 2e-82 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 93.1%

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

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 99.6

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

   4. Applied egg-rr 99.6%

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

Alternative 2: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{z \cdot y - z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{z}}{y - t}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (/ (* x_m 2.0) (- (* z y) (* z t)))))
   (* x_s (if (<= t_1 -5e+101) t_1 (/ (/ (* 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) {
	double t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	double tmp;
	if (t_1 <= -5e+101) {
		tmp = t_1;
	} else {
		tmp = ((x_m * 2.0) / z) / (y - 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / ((z * y) - (z * t))
    if (t_1 <= (-5d+101)) then
        tmp = t_1
    else
        tmp = ((x_m * 2.0d0) / z) / (y - 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 t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	double tmp;
	if (t_1 <= -5e+101) {
		tmp = t_1;
	} else {
		tmp = ((x_m * 2.0) / z) / (y - 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):
	t_1 = (x_m * 2.0) / ((z * y) - (z * t))
	tmp = 0
	if t_1 <= -5e+101:
		tmp = t_1
	else:
		tmp = ((x_m * 2.0) / z) / (y - 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)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(Float64(z * y) - Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -5e+101)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / z) / Float64(y - 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)
	t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	tmp = 0.0;
	if (t_1 <= -5e+101)
		tmp = t_1;
	else
		tmp = ((x_m * 2.0) / z) / (y - 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_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -5e+101], t$95$1, N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.99999999999999989e101

   1. Initial program 97.2%

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

   if -4.99999999999999989e101 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 89.4%

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

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

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. --lowering--.f64 97.7

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

   4. Applied egg-rr 97.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{z \cdot y - z \cdot t} \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{z \cdot y - z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{2}{z}}{y - t}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (/ (* x_m 2.0) (- (* z y) (* z t)))))
   (* x_s (if (<= t_1 -2e-323) t_1 (/ (* 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) {
	double t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	double tmp;
	if (t_1 <= -2e-323) {
		tmp = t_1;
	} else {
		tmp = (x_m * (2.0 / z)) / (y - 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / ((z * y) - (z * t))
    if (t_1 <= (-2d-323)) then
        tmp = t_1
    else
        tmp = (x_m * (2.0d0 / z)) / (y - 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 t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	double tmp;
	if (t_1 <= -2e-323) {
		tmp = t_1;
	} else {
		tmp = (x_m * (2.0 / z)) / (y - 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):
	t_1 = (x_m * 2.0) / ((z * y) - (z * t))
	tmp = 0
	if t_1 <= -2e-323:
		tmp = t_1
	else:
		tmp = (x_m * (2.0 / z)) / (y - 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)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(Float64(z * y) - Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -2e-323)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m * Float64(2.0 / z)) / Float64(y - 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)
	t_1 = (x_m * 2.0) / ((z * y) - (z * t));
	tmp = 0.0;
	if (t_1 <= -2e-323)
		tmp = t_1;
	else
		tmp = (x_m * (2.0 / z)) / (y - 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_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2e-323], t$95$1, N[(N[(x$95$m * N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -1.97626e-323

   1. Initial program 98.5%

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

   if -1.97626e-323 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 86.5%

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

      1. *-commutative N/A

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

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

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 97.5

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{z \cdot y - z \cdot t} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \end{array} \end{array} \]

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 (<= t -7.8e-50)
    (/ (* x_m -2.0) (* z t))
    (if (<= t 2.4e-13) (/ (* x_m 2.0) (* z y)) (* x_m (/ -2.0 (* 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 (t <= -7.8e-50) {
		tmp = (x_m * -2.0) / (z * t);
	} else if (t <= 2.4e-13) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * (-2.0 / (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 (t <= (-7.8d-50)) then
        tmp = (x_m * (-2.0d0)) / (z * t)
    else if (t <= 2.4d-13) then
        tmp = (x_m * 2.0d0) / (z * y)
    else
        tmp = x_m * ((-2.0d0) / (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 (t <= -7.8e-50) {
		tmp = (x_m * -2.0) / (z * t);
	} else if (t <= 2.4e-13) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = x_m * (-2.0 / (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 t <= -7.8e-50:
		tmp = (x_m * -2.0) / (z * t)
	elif t <= 2.4e-13:
		tmp = (x_m * 2.0) / (z * y)
	else:
		tmp = x_m * (-2.0 / (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 (t <= -7.8e-50)
		tmp = Float64(Float64(x_m * -2.0) / Float64(z * t));
	elseif (t <= 2.4e-13)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	else
		tmp = Float64(x_m * Float64(-2.0 / Float64(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 (t <= -7.8e-50)
		tmp = (x_m * -2.0) / (z * t);
	elseif (t <= 2.4e-13)
		tmp = (x_m * 2.0) / (z * y);
	else
		tmp = x_m * (-2.0 / (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[LessEqual[t, -7.8e-50], N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-13], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.80000000000000042e-50

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 84.8

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

   5. Simplified 84.8%

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

   if -7.80000000000000042e-50 < t < 2.3999999999999999e-13

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   if 2.3999999999999999e-13 < t

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 85.3

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

   5. Simplified 85.3%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 85.4

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

   7. Applied egg-rr 85.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \end{array} \end{array} \]

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 (<= t -7.6e-50)
    (/ (* x_m -2.0) (* z t))
    (if (<= t 1.15e-12) (* x_m (/ 2.0 (* z y))) (* x_m (/ -2.0 (* 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 (t <= -7.6e-50) {
		tmp = (x_m * -2.0) / (z * t);
	} else if (t <= 1.15e-12) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = x_m * (-2.0 / (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 (t <= (-7.6d-50)) then
        tmp = (x_m * (-2.0d0)) / (z * t)
    else if (t <= 1.15d-12) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = x_m * ((-2.0d0) / (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 (t <= -7.6e-50) {
		tmp = (x_m * -2.0) / (z * t);
	} else if (t <= 1.15e-12) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = x_m * (-2.0 / (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 t <= -7.6e-50:
		tmp = (x_m * -2.0) / (z * t)
	elif t <= 1.15e-12:
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = x_m * (-2.0 / (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 (t <= -7.6e-50)
		tmp = Float64(Float64(x_m * -2.0) / Float64(z * t));
	elseif (t <= 1.15e-12)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = Float64(x_m * Float64(-2.0 / Float64(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 (t <= -7.6e-50)
		tmp = (x_m * -2.0) / (z * t);
	elseif (t <= 1.15e-12)
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = x_m * (-2.0 / (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[LessEqual[t, -7.6e-50], N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-12], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.5999999999999998e-50

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 84.8

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

   5. Simplified 84.8%

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

   if -7.5999999999999998e-50 < t < 1.14999999999999995e-12

   1. Initial program 90.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 91.2

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 76.0%

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

   if 1.14999999999999995e-12 < t

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 85.3

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

   5. Simplified 85.3%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 85.4

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

   7. Applied egg-rr 85.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{-2}{z \cdot t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (* x_m (/ -2.0 (* z t)))))
   (*
    x_s
    (if (<= t -7.8e-50) t_1 (if (<= t 2.4e-12) (* x_m (/ 2.0 (* z y))) t_1)))))

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 t_1 = x_m * (-2.0 / (z * t));
	double tmp;
	if (t <= -7.8e-50) {
		tmp = t_1;
	} else if (t <= 2.4e-12) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((-2.0d0) / (z * t))
    if (t <= (-7.8d-50)) then
        tmp = t_1
    else if (t <= 2.4d-12) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = t_1
    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 t_1 = x_m * (-2.0 / (z * t));
	double tmp;
	if (t <= -7.8e-50) {
		tmp = t_1;
	} else if (t <= 2.4e-12) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x_m * (-2.0 / (z * t))
	tmp = 0
	if t <= -7.8e-50:
		tmp = t_1
	elif t <= 2.4e-12:
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(-2.0 / Float64(z * t)))
	tmp = 0.0
	if (t <= -7.8e-50)
		tmp = t_1;
	elseif (t <= 2.4e-12)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = t_1;
	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)
	t_1 = x_m * (-2.0 / (z * t));
	tmp = 0.0;
	if (t <= -7.8e-50)
		tmp = t_1;
	elseif (t <= 2.4e-12)
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -7.8e-50], t$95$1, If[LessEqual[t, 2.4e-12], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.80000000000000042e-50 or 2.39999999999999987e-12 < t

   1. Initial program 90.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 85.1

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

   5. Simplified 85.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 85.1

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

   7. Applied egg-rr 85.1%

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

   if -7.80000000000000042e-50 < t < 2.39999999999999987e-12

   1. Initial program 90.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 91.2

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 76.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.0% 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 90.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

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

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 93.2

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

4. Applied egg-rr 93.2%

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

5. Final simplification 93.2%

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

Alternative 8: 53.3% accurate, 1.4× speedup

Math

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

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 (* x_m (/ -2.0 (* 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 * (x_m * (-2.0 / (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 * (x_m * ((-2.0d0) / (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 * (x_m * (-2.0 / (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 * (x_m * (-2.0 / (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(x_m * Float64(-2.0 / 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 * (x_m * (-2.0 / (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[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.6%

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

3. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 62.4

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

5. Simplified 62.4%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 62.3

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

7. Applied egg-rr 62.3%

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

8. Final simplification 62.3%

   \[\leadsto x \cdot \frac{-2}{z \cdot t} \]
9. 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 2024198 
(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))))