Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 83.7% → 96.9%

Time: 7.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{t - z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * (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 * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{t - z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * (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 * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.8e+27)
    (/ (* (- y z) x_m) (- t z))
    (/ (- z y) (/ (- z t) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.8e+27) {
		tmp = ((y - z) * x_m) / (t - z);
	} else {
		tmp = (z - y) / ((z - t) / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 1.8d+27) then
        tmp = ((y - z) * x_m) / (t - z)
    else
        tmp = (z - y) / ((z - t) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.8e+27) {
		tmp = ((y - z) * x_m) / (t - z);
	} else {
		tmp = (z - y) / ((z - t) / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 1.8e+27:
		tmp = ((y - z) * x_m) / (t - z)
	else:
		tmp = (z - y) / ((z - t) / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 1.8e+27)
		tmp = Float64(Float64(Float64(y - z) * x_m) / Float64(t - z));
	else
		tmp = Float64(Float64(z - y) / Float64(Float64(z - t) / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 1.8e+27)
		tmp = ((y - z) * x_m) / (t - z);
	else
		tmp = (z - y) / ((z - t) / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 1.8e+27], N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(z - y), $MachinePrecision] / N[(N[(z - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999991e27

   1. Initial program 91.9%

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

   if 1.79999999999999991e27 < x

   1. Initial program 68.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. associate-/r/ N/A

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

      6. un-div-inv N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. lift-neg.f64 N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 96.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\_m\\ \mathbf{elif}\;y \leq -1.26:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, x\_m, x\_m\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -1.02e+121)
    (* (/ y (- t z)) x_m)
    (if (<= y -1.26)
      (fma (/ (- y) z) x_m x_m)
      (if (<= y 3.1e+28) (* (/ z (- z t)) x_m) (* (/ x_m (- t z)) y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.02e+121) {
		tmp = (y / (t - z)) * x_m;
	} else if (y <= -1.26) {
		tmp = fma((-y / z), x_m, x_m);
	} else if (y <= 3.1e+28) {
		tmp = (z / (z - t)) * x_m;
	} else {
		tmp = (x_m / (t - z)) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.02e+121)
		tmp = Float64(Float64(y / Float64(t - z)) * x_m);
	elseif (y <= -1.26)
		tmp = fma(Float64(Float64(-y) / z), x_m, x_m);
	elseif (y <= 3.1e+28)
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.02e+121], N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[y, -1.26], N[(N[((-y) / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], If[LessEqual[y, 3.1e+28], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < -1.02000000000000005e121

   1. Initial program 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if -1.02000000000000005e121 < y < -1.26000000000000001

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 60.5

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - y}{z}, x, x\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{z}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.9%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, x, x\right) \]

   if -1.26000000000000001 < y < 3.1000000000000001e28

   1. Initial program 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - z\right)}}{z - y}} \cdot x \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - z\right)}}{z - y}} \cdot x \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(z\right)\right)\right)}}{z - y}} \cdot x \]

      17. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \left(t + \color{blue}{\left(-z\right)}\right)}{z - y}} \cdot x \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(-z\right) + t\right)}}{z - y}} \cdot x \]

      19. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(-z\right)\right) - t}}{z - y}} \cdot x \]

      20. neg-sub0 N/A

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

      21. lift-neg.f64 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 N/A

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

      24. lower--.f64 94.7

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

   6. Applied rewrites 94.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 79.3

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

   9. Applied rewrites 79.3%

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

   if 3.1000000000000001e28 < y

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 3: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;\frac{-x\_m}{z} \cdot y\\ \mathbf{elif}\;z \leq 125:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.85e+111)
    (* 1.0 x_m)
    (if (<= z -1.45e-57)
      (* (/ (- x_m) z) y)
      (if (<= z 125.0) (* (/ x_m t) y) (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.85e+111) {
		tmp = 1.0 * x_m;
	} else if (z <= -1.45e-57) {
		tmp = (-x_m / z) * y;
	} else if (z <= 125.0) {
		tmp = (x_m / t) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.85d+111)) then
        tmp = 1.0d0 * x_m
    else if (z <= (-1.45d-57)) then
        tmp = (-x_m / z) * y
    else if (z <= 125.0d0) then
        tmp = (x_m / t) * y
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.85e+111) {
		tmp = 1.0 * x_m;
	} else if (z <= -1.45e-57) {
		tmp = (-x_m / z) * y;
	} else if (z <= 125.0) {
		tmp = (x_m / t) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.85e+111:
		tmp = 1.0 * x_m
	elif z <= -1.45e-57:
		tmp = (-x_m / z) * y
	elif z <= 125.0:
		tmp = (x_m / t) * y
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.85e+111)
		tmp = Float64(1.0 * x_m);
	elseif (z <= -1.45e-57)
		tmp = Float64(Float64(Float64(-x_m) / z) * y);
	elseif (z <= 125.0)
		tmp = Float64(Float64(x_m / t) * y);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.85e+111)
		tmp = 1.0 * x_m;
	elseif (z <= -1.45e-57)
		tmp = (-x_m / z) * y;
	elseif (z <= 125.0)
		tmp = (x_m / t) * y;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.85e+111], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, -1.45e-57], N[(N[((-x$95$m) / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 125.0], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.8500000000000001e111 or 125 < z

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.8500000000000001e111 < z < -1.45000000000000013e-57

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \frac{x}{z}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

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

   if -1.45000000000000013e-57 < z < 125

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 58.7%

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

Alternative 4: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, x\_m, x\_m\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+182}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.8e+98)
    (fma (/ (- y) z) x_m x_m)
    (if (<= z 2.1e+182) (* (/ x_m (- t z)) (- y z)) (* (/ z (- z t)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+98) {
		tmp = fma((-y / z), x_m, x_m);
	} else if (z <= 2.1e+182) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+98)
		tmp = fma(Float64(Float64(-y) / z), x_m, x_m);
	elseif (z <= 2.1e+182)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.8e+98], N[(N[((-y) / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], If[LessEqual[z, 2.1e+182], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.7999999999999999e98

   1. Initial program 69.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 16.0

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - y}{z}, x, x\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{z}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.2%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, x, x\right) \]

   if -3.7999999999999999e98 < z < 2.0999999999999999e182

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 2.0999999999999999e182 < z

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - z\right)}}{z - y}} \cdot x \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - z\right)}}{z - y}} \cdot x \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(z\right)\right)\right)}}{z - y}} \cdot x \]

      17. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \left(t + \color{blue}{\left(-z\right)}\right)}{z - y}} \cdot x \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(-z\right) + t\right)}}{z - y}} \cdot x \]

      19. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(-z\right)\right) - t}}{z - y}} \cdot x \]

      20. neg-sub0 N/A

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

      21. lift-neg.f64 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.6

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

   9. Applied rewrites 95.6%

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

Alternative 5: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t - z} \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ x_m (- t z)) y)))
   (*
    x_s
    (if (<= y -7.8e+36) t_1 (if (<= y 3.1e+28) (* (/ z (- z t)) x_m) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / (t - z)) * y;
	double tmp;
	if (y <= -7.8e+36) {
		tmp = t_1;
	} else if (y <= 3.1e+28) {
		tmp = (z / (z - t)) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / (t - z)) * y
    if (y <= (-7.8d+36)) then
        tmp = t_1
    else if (y <= 3.1d+28) then
        tmp = (z / (z - t)) * x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / (t - z)) * y;
	double tmp;
	if (y <= -7.8e+36) {
		tmp = t_1;
	} else if (y <= 3.1e+28) {
		tmp = (z / (z - t)) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / (t - z)) * y
	tmp = 0
	if y <= -7.8e+36:
		tmp = t_1
	elif y <= 3.1e+28:
		tmp = (z / (z - t)) * x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / Float64(t - z)) * y)
	tmp = 0.0
	if (y <= -7.8e+36)
		tmp = t_1;
	elseif (y <= 3.1e+28)
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / (t - z)) * y;
	tmp = 0.0;
	if (y <= -7.8e+36)
		tmp = t_1;
	elseif (y <= 3.1e+28)
		tmp = (z / (z - t)) * x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -7.8e+36], t$95$1, If[LessEqual[y, 3.1e+28], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000042e36 or 3.1000000000000001e28 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if -7.80000000000000042e36 < y < 3.1000000000000001e28

   1. Initial program 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - z\right)}}{z - y}} \cdot x \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - z\right)}}{z - y}} \cdot x \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(z\right)\right)\right)}}{z - y}} \cdot x \]

      17. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{0 - \left(t + \color{blue}{\left(-z\right)}\right)}{z - y}} \cdot x \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(-z\right) + t\right)}}{z - y}} \cdot x \]

      19. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(-z\right)\right) - t}}{z - y}} \cdot x \]

      20. neg-sub0 N/A

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

      21. lift-neg.f64 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 N/A

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

      24. lower--.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 77.8

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

   9. Applied rewrites 77.8%

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

Alternative 6: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3050000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) x_m) t)))
   (*
    x_s
    (if (<= t -3050000000000.0)
      t_1
      (if (<= t 3.5e+94) (fma (/ (- y) z) x_m x_m) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((y - z) * x_m) / t;
	double tmp;
	if (t <= -3050000000000.0) {
		tmp = t_1;
	} else if (t <= 3.5e+94) {
		tmp = fma((-y / z), x_m, x_m);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(Float64(y - z) * x_m) / t)
	tmp = 0.0
	if (t <= -3050000000000.0)
		tmp = t_1;
	elseif (t <= 3.5e+94)
		tmp = fma(Float64(Float64(-y) / z), x_m, x_m);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -3050000000000.0], t$95$1, If[LessEqual[t, 3.5e+94], N[(N[((-y) / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.05e12 or 3.4999999999999997e94 < t

   1. Initial program 86.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if -3.05e12 < t < 3.4999999999999997e94

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 48.5

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - y}{z}, x, x\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{z}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.6%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2900000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;x\_m - \frac{y \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) x_m) t)))
   (*
    x_s
    (if (<= t -2900000000000.0)
      t_1
      (if (<= t 3.5e+94) (- x_m (/ (* y x_m) z)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((y - z) * x_m) / t;
	double tmp;
	if (t <= -2900000000000.0) {
		tmp = t_1;
	} else if (t <= 3.5e+94) {
		tmp = x_m - ((y * x_m) / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y - z) * x_m) / t
    if (t <= (-2900000000000.0d0)) then
        tmp = t_1
    else if (t <= 3.5d+94) then
        tmp = x_m - ((y * x_m) / z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((y - z) * x_m) / t;
	double tmp;
	if (t <= -2900000000000.0) {
		tmp = t_1;
	} else if (t <= 3.5e+94) {
		tmp = x_m - ((y * x_m) / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = ((y - z) * x_m) / t
	tmp = 0
	if t <= -2900000000000.0:
		tmp = t_1
	elif t <= 3.5e+94:
		tmp = x_m - ((y * x_m) / z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(Float64(y - z) * x_m) / t)
	tmp = 0.0
	if (t <= -2900000000000.0)
		tmp = t_1;
	elseif (t <= 3.5e+94)
		tmp = Float64(x_m - Float64(Float64(y * x_m) / z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = ((y - z) * x_m) / t;
	tmp = 0.0;
	if (t <= -2900000000000.0)
		tmp = t_1;
	elseif (t <= 3.5e+94)
		tmp = x_m - ((y * x_m) / z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2900000000000.0], t$95$1, If[LessEqual[t, 3.5e+94], N[(x$95$m - N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e12 or 3.4999999999999997e94 < t

   1. Initial program 86.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if -2.9e12 < t < 3.4999999999999997e94

   1. Initial program 87.8%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

Alternative 8: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - \frac{y \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -550000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (- x_m (/ (* y x_m) z))))
   (*
    x_s
    (if (<= z -550000000.0)
      t_1
      (if (<= z 7.5e-40) (* (/ x_m (- t z)) y) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m - ((y * x_m) / z);
	double tmp;
	if (z <= -550000000.0) {
		tmp = t_1;
	} else if (z <= 7.5e-40) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m - ((y * x_m) / z)
    if (z <= (-550000000.0d0)) then
        tmp = t_1
    else if (z <= 7.5d-40) then
        tmp = (x_m / (t - z)) * y
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m - ((y * x_m) / z);
	double tmp;
	if (z <= -550000000.0) {
		tmp = t_1;
	} else if (z <= 7.5e-40) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m - ((y * x_m) / z)
	tmp = 0
	if z <= -550000000.0:
		tmp = t_1
	elif z <= 7.5e-40:
		tmp = (x_m / (t - z)) * y
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m - Float64(Float64(y * x_m) / z))
	tmp = 0.0
	if (z <= -550000000.0)
		tmp = t_1;
	elseif (z <= 7.5e-40)
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m - ((y * x_m) / z);
	tmp = 0.0;
	if (z <= -550000000.0)
		tmp = t_1;
	elseif (z <= 7.5e-40)
		tmp = (x_m / (t - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m - N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -550000000.0], t$95$1, If[LessEqual[z, 7.5e-40], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e8 or 7.50000000000000069e-40 < z

   1. Initial program 81.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -5.5e8 < z < 7.50000000000000069e-40

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 9: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -2e+111)
    (* 1.0 x_m)
    (if (<= z 1.2e+123) (* (/ x_m (- t z)) y) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2e+111) {
		tmp = 1.0 * x_m;
	} else if (z <= 1.2e+123) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2d+111)) then
        tmp = 1.0d0 * x_m
    else if (z <= 1.2d+123) then
        tmp = (x_m / (t - z)) * y
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2e+111) {
		tmp = 1.0 * x_m;
	} else if (z <= 1.2e+123) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -2e+111:
		tmp = 1.0 * x_m
	elif z <= 1.2e+123:
		tmp = (x_m / (t - z)) * y
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2e+111)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 1.2e+123)
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -2e+111)
		tmp = 1.0 * x_m;
	elseif (z <= 1.2e+123)
		tmp = (x_m / (t - z)) * y;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2e+111], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 1.2e+123], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999991e111 or 1.19999999999999994e123 < z

   1. Initial program 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.99999999999999991e111 < z < 1.19999999999999994e123

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

Alternative 10: 61.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1500:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 235:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1500.0)
    (* 1.0 x_m)
    (if (<= z 235.0) (* (/ y t) x_m) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1500.0) {
		tmp = 1.0 * x_m;
	} else if (z <= 235.0) {
		tmp = (y / t) * x_m;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1500.0d0)) then
        tmp = 1.0d0 * x_m
    else if (z <= 235.0d0) then
        tmp = (y / t) * x_m
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1500.0) {
		tmp = 1.0 * x_m;
	} else if (z <= 235.0) {
		tmp = (y / t) * x_m;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1500.0:
		tmp = 1.0 * x_m
	elif z <= 235.0:
		tmp = (y / t) * x_m
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1500.0)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 235.0)
		tmp = Float64(Float64(y / t) * x_m);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1500.0)
		tmp = 1.0 * x_m;
	elseif (z <= 235.0)
		tmp = (y / t) * x_m;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1500.0], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 235.0], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1500 or 235 < z

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 57.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1500 < z < 235

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

Alternative 11: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1500:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 125:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1500.0)
    (* 1.0 x_m)
    (if (<= z 125.0) (* (/ x_m t) y) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1500.0) {
		tmp = 1.0 * x_m;
	} else if (z <= 125.0) {
		tmp = (x_m / t) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1500.0d0)) then
        tmp = 1.0d0 * x_m
    else if (z <= 125.0d0) then
        tmp = (x_m / t) * y
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1500.0) {
		tmp = 1.0 * x_m;
	} else if (z <= 125.0) {
		tmp = (x_m / t) * y;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1500.0:
		tmp = 1.0 * x_m
	elif z <= 125.0:
		tmp = (x_m / t) * y
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1500.0)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 125.0)
		tmp = Float64(Float64(x_m / t) * y);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1500.0)
		tmp = 1.0 * x_m;
	elseif (z <= 125.0)
		tmp = (x_m / t) * y;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1500.0], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 125.0], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1500 or 125 < z

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 57.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1500 < z < 125

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 56.5%

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

Alternative 12: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.9

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

4. Applied rewrites 95.9%

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

Alternative 13: 35.7% accurate, 3.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* 1.0 x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.9

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

4. Applied rewrites 95.9%

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

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{1} \cdot x \]
6. Step-by-step derivation

7. Applied rewrites 34.5%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 96.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x / ((t - z) / (y - 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 / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ (- t z) (- y z))))

  (/ (* x (- y z)) (- t z)))