Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.2% → 98.3%

Time: 7.7s

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: 89.2% 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: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{y - t} \cdot \frac{2}{z\_m}\\ t_2 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+236}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x (- y t)) (/ 2.0 z_m))) (t_2 (- (* y z_m) (* t z_m))))
   (*
    z_s
    (if (<= t_2 (- INFINITY))
      t_1
      (if (<= t_2 1e+236) (/ (* 2.0 x) (* (- y t) z_m)) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (y - t)) * (2.0 / z_m);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+236) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (y - t)) * (2.0 / z_m);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 1e+236) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / (y - t)) * (2.0 / z_m)
	t_2 = (y * z_m) - (t * z_m)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 1e+236:
		tmp = (2.0 * x) / ((y - t) * z_m)
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / z_m))
	t_2 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+236)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / (y - t)) * (2.0 / z_m);
	t_2 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 1e+236)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+236], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0 or 1.00000000000000005e236 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 75.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.00000000000000005e236

   1. Initial program 98.2%

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

      1. lift--.f64 N/A

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \leq -\infty:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \leq 10^{+236}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-33}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot x}{z\_m}}{t - y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 8e-33)
    (/ (* 2.0 x) (* (- y t) z_m))
    (/ (/ (* -2.0 x) z_m) (- t y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 8e-33) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((-2.0 * x) / z_m) / (t - y);
	}
	return z_s * tmp;
}

Fortran

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

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 8e-33) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((-2.0 * x) / z_m) / (t - y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 8e-33:
		tmp = (2.0 * x) / ((y - t) * z_m)
	else:
		tmp = ((-2.0 * x) / z_m) / (t - y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 8e-33)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(-2.0 * x) / z_m) / Float64(t - y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 8e-33)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = ((-2.0 * x) / z_m) / (t - y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 8e-33], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * x), $MachinePrecision] / z$95$m), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.0000000000000004e-33

   1. Initial program 94.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 94.8

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

   4. Applied rewrites 94.8%

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

   if 8.0000000000000004e-33 < z

   1. Initial program 85.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-frac-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. distribute-lft-neg-in N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. neg-sub0 N/A

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

      24. lift--.f64 N/A

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

      25. sub-neg N/A

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

      26. +-commutative N/A

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

   6. Applied rewrites 97.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-33}:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot x}{z}}{t - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 33000000:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m} \cdot x}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 33000000.0)
    (/ (* 2.0 x) (* (- y t) z_m))
    (/ (* (/ 2.0 z_m) x) (- y t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 33000000.0) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 33000000.0d0) then
        tmp = (2.0d0 * x) / ((y - t) * z_m)
    else
        tmp = ((2.0d0 / z_m) * x) / (y - t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 33000000.0) {
		tmp = (2.0 * x) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 33000000.0:
		tmp = (2.0 * x) / ((y - t) * z_m)
	else:
		tmp = ((2.0 / z_m) * x) / (y - t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 33000000.0)
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(2.0 / z_m) * x) / Float64(y - t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 33000000.0)
		tmp = (2.0 * x) / ((y - t) * z_m);
	else
		tmp = ((2.0 / z_m) * x) / (y - t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 33000000.0], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / z$95$m), $MachinePrecision] * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.3e7

   1. Initial program 95.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 95.1

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

   4. Applied rewrites 95.1%

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

   if 3.3e7 < z

   1. Initial program 83.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 97.1

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

   4. Applied rewrites 97.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 33000000:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z} \cdot x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{elif}\;t \leq 3.45 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y \cdot z\_m} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot z\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1.12e-100)
    (* (/ x (* t z_m)) -2.0)
    (if (<= t 3.45e+99) (* (/ x (* y z_m)) 2.0) (* (/ -2.0 (* t z_m)) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.12e-100) {
		tmp = (x / (t * z_m)) * -2.0;
	} else if (t <= 3.45e+99) {
		tmp = (x / (y * z_m)) * 2.0;
	} else {
		tmp = (-2.0 / (t * z_m)) * x;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.12d-100)) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else if (t <= 3.45d+99) then
        tmp = (x / (y * z_m)) * 2.0d0
    else
        tmp = ((-2.0d0) / (t * z_m)) * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.12e-100) {
		tmp = (x / (t * z_m)) * -2.0;
	} else if (t <= 3.45e+99) {
		tmp = (x / (y * z_m)) * 2.0;
	} else {
		tmp = (-2.0 / (t * z_m)) * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -1.12e-100:
		tmp = (x / (t * z_m)) * -2.0
	elif t <= 3.45e+99:
		tmp = (x / (y * z_m)) * 2.0
	else:
		tmp = (-2.0 / (t * z_m)) * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -1.12e-100)
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	elseif (t <= 3.45e+99)
		tmp = Float64(Float64(x / Float64(y * z_m)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 / Float64(t * z_m)) * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.12e-100)
		tmp = (x / (t * z_m)) * -2.0;
	elseif (t <= 3.45e+99)
		tmp = (x / (y * z_m)) * 2.0;
	else
		tmp = (-2.0 / (t * z_m)) * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -1.12e-100], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t, 3.45e+99], N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.11999999999999996e-100

   1. Initial program 90.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-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 70.3

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

   5. Applied rewrites 70.3%

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

   if -1.11999999999999996e-100 < t < 3.45e99

   1. Initial program 93.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 3.45e99 < t

   1. Initial program 90.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-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 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 85.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{elif}\;t \leq 3.45 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * ((2.0 * x) / ((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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(2.0 * x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 92.9

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

4. Applied rewrites 92.9%

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

5. Final simplification 92.9%

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

Alternative 6: 91.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 94.3

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

4. Applied rewrites 94.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 92.8%

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

Alternative 7: 52.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. *-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 52.5

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

5. Applied rewrites 52.5%

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

Alternative 8: 52.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. *-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 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 52.5%

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

8. Final simplification 52.5%

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

Developer Target 1: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024331 
(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))))