Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 98.1%

Time: 8.8s

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: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2 \cdot x\_m}{y \cdot z\_m - t \cdot z\_m}\\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{2}{y - t} \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x\_m}{\left(y - t\right) \cdot z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 x_m) (- (* y z_m) (* t z_m)))))
   (*
    x_s
    (*
     z_s
     (if (<= t_1 -5e-317)
       t_1
       (if (<= t_1 0.0)
         (/ (* (/ 2.0 (- y t)) x_m) z_m)
         (/ (* 2.0 x_m) (* (- y t) z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	double tmp;
	if (t_1 <= -5e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((2.0 / (y - t)) * x_m) / z_m;
	} else {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * x_m) / ((y * z_m) - (t * z_m))
    if (t_1 <= (-5d-317)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = ((2.0d0 / (y - t)) * x_m) / z_m
    else
        tmp = (2.0d0 * x_m) / ((y - t) * z_m)
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	double tmp;
	if (t_1 <= -5e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((2.0 / (y - t)) * x_m) / z_m;
	} else {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m))
	tmp = 0
	if t_1 <= -5e-317:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = ((2.0 / (y - t)) * x_m) / z_m
	else:
		tmp = (2.0 * x_m) / ((y - t) * z_m)
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(2.0 * x_m) / Float64(Float64(y * z_m) - Float64(t * z_m)))
	tmp = 0.0
	if (t_1 <= -5e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(y - t)) * x_m) / z_m);
	else
		tmp = Float64(Float64(2.0 * x_m) / Float64(Float64(y - t) * z_m));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	tmp = 0.0;
	if (t_1 <= -5e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = ((2.0 / (y - t)) * x_m) / z_m;
	else
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(z$95$s * If[LessEqual[t$95$1, -5e-317], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -5.00000017e-317

   1. Initial program 96.6%

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

   if -5.00000017e-317 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < 0.0

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 0.0 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 88.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot x}{y \cdot z - t \cdot z} \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;\frac{2 \cdot x}{y \cdot z - t \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{2}{y - t} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2 \cdot x\_m}{y \cdot z\_m - t \cdot z\_m}\\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{2}{\frac{y - t}{x\_m} \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x\_m}{\left(y - t\right) \cdot z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 x_m) (- (* y z_m) (* t z_m)))))
   (*
    x_s
    (*
     z_s
     (if (<= t_1 -5e-317)
       t_1
       (if (<= t_1 0.0)
         (/ 2.0 (* (/ (- y t) x_m) z_m))
         (/ (* 2.0 x_m) (* (- y t) z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	double tmp;
	if (t_1 <= -5e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / (((y - t) / x_m) * z_m);
	} else {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * x_m) / ((y * z_m) - (t * z_m))
    if (t_1 <= (-5d-317)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 / (((y - t) / x_m) * z_m)
    else
        tmp = (2.0d0 * x_m) / ((y - t) * z_m)
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	double tmp;
	if (t_1 <= -5e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / (((y - t) / x_m) * z_m);
	} else {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m))
	tmp = 0
	if t_1 <= -5e-317:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = 2.0 / (((y - t) / x_m) * z_m)
	else:
		tmp = (2.0 * x_m) / ((y - t) * z_m)
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(2.0 * x_m) / Float64(Float64(y * z_m) - Float64(t * z_m)))
	tmp = 0.0
	if (t_1 <= -5e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(y - t) / x_m) * z_m));
	else
		tmp = Float64(Float64(2.0 * x_m) / Float64(Float64(y - t) * z_m));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	t_1 = (2.0 * x_m) / ((y * z_m) - (t * z_m));
	tmp = 0.0;
	if (t_1 <= -5e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = 2.0 / (((y - t) / x_m) * z_m);
	else
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(z$95$s * If[LessEqual[t$95$1, -5e-317], t$95$1, If[LessEqual[t$95$1, 0.0], N[(2.0 / N[(N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -5.00000017e-317

   1. Initial program 96.6%

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

   if -5.00000017e-317 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < 0.0

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. metadata-eval N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 98.6

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if 0.0 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 88.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot x}{y \cdot z - t \cdot z} \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;\frac{2 \cdot x}{y \cdot z - t \cdot z} \leq 0:\\ \;\;\;\;\frac{2}{\frac{y - t}{x} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 \cdot x\_m}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m}}{0.5 \cdot \left(y - t\right)}\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= z_m 1.5e-58)
     (/ (* 2.0 x_m) (* (- y t) z_m))
     (/ (/ x_m z_m) (* 0.5 (- y t)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.5e-58) {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	} else {
		tmp = (x_m / z_m) / (0.5 * (y - t));
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 1.5d-58) then
        tmp = (2.0d0 * x_m) / ((y - t) * z_m)
    else
        tmp = (x_m / z_m) / (0.5d0 * (y - t))
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.5e-58) {
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	} else {
		tmp = (x_m / z_m) / (0.5 * (y - t));
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 1.5e-58:
		tmp = (2.0 * x_m) / ((y - t) * z_m)
	else:
		tmp = (x_m / z_m) / (0.5 * (y - t))
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.5e-58)
		tmp = Float64(Float64(2.0 * x_m) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(x_m / z_m) / Float64(0.5 * Float64(y - t)));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 1.5e-58)
		tmp = (2.0 * x_m) / ((y - t) * z_m);
	else
		tmp = (x_m / z_m) / (0.5 * (y - t));
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.5e-58], N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(0.5 * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.50000000000000004e-58

   1. Initial program 91.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 94.5

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

   4. Applied rewrites 94.5%

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

   if 1.50000000000000004e-58 < z

   1. Initial program 84.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 97.3

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

   4. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\left(y - t\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{0.5 \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{x\_m}{t \cdot z\_m}\\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{2 \cdot x\_m}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* -2.0 (/ x_m (* t z_m)))))
   (*
    x_s
    (*
     z_s
     (if (<= t -1.1e-52)
       t_1
       (if (<= t 7e+59) (/ (* 2.0 x_m) (* y z_m)) t_1))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (t * z_m));
	double tmp;
	if (t <= -1.1e-52) {
		tmp = t_1;
	} else if (t <= 7e+59) {
		tmp = (2.0 * x_m) / (y * z_m);
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-2.0d0) * (x_m / (t * z_m))
    if (t <= (-1.1d-52)) then
        tmp = t_1
    else if (t <= 7d+59) then
        tmp = (2.0d0 * x_m) / (y * z_m)
    else
        tmp = t_1
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (t * z_m));
	double tmp;
	if (t <= -1.1e-52) {
		tmp = t_1;
	} else if (t <= 7e+59) {
		tmp = (2.0 * x_m) / (y * z_m);
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	t_1 = -2.0 * (x_m / (t * z_m))
	tmp = 0
	if t <= -1.1e-52:
		tmp = t_1
	elif t <= 7e+59:
		tmp = (2.0 * x_m) / (y * z_m)
	else:
		tmp = t_1
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	t_1 = Float64(-2.0 * Float64(x_m / Float64(t * z_m)))
	tmp = 0.0
	if (t <= -1.1e-52)
		tmp = t_1;
	elseif (t <= 7e+59)
		tmp = Float64(Float64(2.0 * x_m) / Float64(y * z_m));
	else
		tmp = t_1;
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	t_1 = -2.0 * (x_m / (t * z_m));
	tmp = 0.0;
	if (t <= -1.1e-52)
		tmp = t_1;
	elseif (t <= 7e+59)
		tmp = (2.0 * x_m) / (y * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(-2.0 * N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(z$95$s * If[LessEqual[t, -1.1e-52], t$95$1, If[LessEqual[t, 7e+59], N[(N[(2.0 * x$95$m), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.10000000000000005e-52 or 7e59 < t

   1. Initial program 86.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -1.10000000000000005e-52 < t < 7e59

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{x\_m}{t \cdot z\_m}\\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{y \cdot z\_m} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* -2.0 (/ x_m (* t z_m)))))
   (*
    x_s
    (*
     z_s
     (if (<= t -1.1e-52)
       t_1
       (if (<= t 7e+59) (* (/ 2.0 (* y z_m)) x_m) t_1))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (t * z_m));
	double tmp;
	if (t <= -1.1e-52) {
		tmp = t_1;
	} else if (t <= 7e+59) {
		tmp = (2.0 / (y * z_m)) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-2.0d0) * (x_m / (t * z_m))
    if (t <= (-1.1d-52)) then
        tmp = t_1
    else if (t <= 7d+59) then
        tmp = (2.0d0 / (y * z_m)) * x_m
    else
        tmp = t_1
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (t * z_m));
	double tmp;
	if (t <= -1.1e-52) {
		tmp = t_1;
	} else if (t <= 7e+59) {
		tmp = (2.0 / (y * z_m)) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	t_1 = -2.0 * (x_m / (t * z_m))
	tmp = 0
	if t <= -1.1e-52:
		tmp = t_1
	elif t <= 7e+59:
		tmp = (2.0 / (y * z_m)) * x_m
	else:
		tmp = t_1
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	t_1 = Float64(-2.0 * Float64(x_m / Float64(t * z_m)))
	tmp = 0.0
	if (t <= -1.1e-52)
		tmp = t_1;
	elseif (t <= 7e+59)
		tmp = Float64(Float64(2.0 / Float64(y * z_m)) * x_m);
	else
		tmp = t_1;
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	t_1 = -2.0 * (x_m / (t * z_m));
	tmp = 0.0;
	if (t <= -1.1e-52)
		tmp = t_1;
	elseif (t <= 7e+59)
		tmp = (2.0 / (y * z_m)) * x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(-2.0 * N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(z$95$s * If[LessEqual[t, -1.1e-52], t$95$1, If[LessEqual[t, 7e+59], N[(N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.10000000000000005e-52 or 7e59 < t

   1. Initial program 86.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -1.10000000000000005e-52 < t < 7e59

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{y \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 93.6

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

4. Applied rewrites 93.6%

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

5. Final simplification 93.6%

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

Alternative 7: 54.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 50.5

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

5. Applied rewrites 50.5%

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

6. Final simplification 50.5%

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

Alternative 8: 54.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 50.5

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

5. Applied rewrites 50.5%

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

7. Applied rewrites 50.5%

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

Developer Target 1: 97.1% 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 2024255 
(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))))