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

Percentage Accurate: 84.6% → 96.8%

Time: 8.6s

Alternatives: 14

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: 84.6% 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: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t + z} \cdot \left(\left(y - z\right) \cdot \frac{t + z}{t - z}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.7e-18)
    (/ (* x_m (- y z)) (- t z))
    (* (/ x_m (+ t z)) (* (- y z) (/ (+ t z) (- t z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.7e-18) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t + z)) * ((y - z) * ((t + z) / (t - z)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2.7d-18) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (x_m / (t + z)) * ((y - z) * ((t + z) / (t - z)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.7e-18) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t + z)) * ((y - z) * ((t + z) / (t - z)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2.7e-18:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (t + z)) * ((y - z) * ((t + z) / (t - z)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2.7e-18)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(t + z)) * Float64(Float64(y - z) * Float64(Float64(t + z) / Float64(t - z))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2.7e-18)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (x_m / (t + z)) * ((y - z) * ((t + z) / (t - z)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2.7e-18], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t + z), $MachinePrecision]), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] * N[(N[(t + z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.69999999999999989e-18

   1. Initial program 90.1%

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

   if 2.69999999999999989e-18 < x

   1. Initial program 74.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. un-div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 98.2%

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

Alternative 2: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, t, x\_m\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-89} \lor \neg \left(z \leq 2.16 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x\_m}{z - t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \end{array} \end{array} \]

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 (<= z -1.26e+124)
    (fma (/ x_m z) t x_m)
    (if (or (<= z -6e-89) (not (<= z 2.16e-32)))
      (* (/ x_m (- z t)) z)
      (* (/ x_m t) 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 (z <= -1.26e+124) {
		tmp = fma((x_m / z), t, x_m);
	} else if ((z <= -6e-89) || !(z <= 2.16e-32)) {
		tmp = (x_m / (z - t)) * z;
	} else {
		tmp = (x_m / t) * 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 (z <= -1.26e+124)
		tmp = fma(Float64(x_m / z), t, x_m);
	elseif ((z <= -6e-89) || !(z <= 2.16e-32))
		tmp = Float64(Float64(x_m / Float64(z - t)) * z);
	else
		tmp = Float64(Float64(x_m / t) * 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[z, -1.26e+124], N[(N[(x$95$m / z), $MachinePrecision] * t + x$95$m), $MachinePrecision], If[Or[LessEqual[z, -6e-89], N[Not[LessEqual[z, 2.16e-32]], $MachinePrecision]], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.26e124

   1. Initial program 63.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 81.3%

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

   if -1.26e124 < z < -5.9999999999999999e-89 or 2.1600000000000001e-32 < z

   1. Initial program 86.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 62.6

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 62.3%

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

   if -5.9999999999999999e-89 < z < 2.1600000000000001e-32

   1. Initial program 93.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 75.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-89} \lor \neg \left(z \leq 2.16 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{z - t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{-z}, x\_m, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-32}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (fma (/ y (- z)) x_m x_m)))
   (*
    x_s
    (if (<= z -6.2e+133)
      t_1
      (if (<= z -1.06e-32)
        (/ (* z x_m) (- z t))
        (if (<= z 4.4e-27) (* (/ 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 = fma((y / -z), x_m, x_m);
	double tmp;
	if (z <= -6.2e+133) {
		tmp = t_1;
	} else if (z <= -1.06e-32) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 4.4e-27) {
		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 = fma(Float64(y / Float64(-z)), x_m, x_m)
	tmp = 0.0
	if (z <= -6.2e+133)
		tmp = t_1;
	elseif (z <= -1.06e-32)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	elseif (z <= 4.4e-27)
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	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[(y / (-z)), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -6.2e+133], t$95$1, If[LessEqual[z, -1.06e-32], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-27], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2e133 or 4.39999999999999974e-27 < z

   1. Initial program 70.8%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 75.2%

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

   if -6.2e133 < z < -1.05999999999999994e-32

   1. Initial program 97.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -1.05999999999999994e-32 < z < 4.39999999999999974e-27

   1. Initial program 94.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-32}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-32}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot z\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9e+173)
    (/ x_m 1.0)
    (if (<= z -1.06e-32)
      (/ (* z x_m) (- z t))
      (if (<= z 3.8e-28) (* (/ x_m (- t z)) y) (* (/ x_m (- z t)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -1.06e-32) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 3.8e-28) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9d+173)) then
        tmp = x_m / 1.0d0
    else if (z <= (-1.06d-32)) then
        tmp = (z * x_m) / (z - t)
    else if (z <= 3.8d-28) then
        tmp = (x_m / (t - z)) * y
    else
        tmp = (x_m / (z - t)) * z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -1.06e-32) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 3.8e-28) {
		tmp = (x_m / (t - z)) * y;
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9e+173:
		tmp = x_m / 1.0
	elif z <= -1.06e-32:
		tmp = (z * x_m) / (z - t)
	elif z <= 3.8e-28:
		tmp = (x_m / (t - z)) * y
	else:
		tmp = (x_m / (z - t)) * z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9e+173)
		tmp = Float64(x_m / 1.0);
	elseif (z <= -1.06e-32)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	elseif (z <= 3.8e-28)
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9e+173)
		tmp = x_m / 1.0;
	elseif (z <= -1.06e-32)
		tmp = (z * x_m) / (z - t);
	elseif (z <= 3.8e-28)
		tmp = (x_m / (t - z)) * y;
	else
		tmp = (x_m / (z - t)) * z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9e+173], N[(x$95$m / 1.0), $MachinePrecision], If[LessEqual[z, -1.06e-32], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-28], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -9.0000000000000004e173

   1. Initial program 56.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 88.8%

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

   if -9.0000000000000004e173 < z < -1.05999999999999994e-32

   1. Initial program 93.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 70.7

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

   7. Applied rewrites 70.7%

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

   if -1.05999999999999994e-32 < z < 3.80000000000000009e-28

   1. Initial program 93.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   if 3.80000000000000009e-28 < z

   1. Initial program 77.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 61.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-32}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-28}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot z\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9e+173)
    (/ x_m 1.0)
    (if (<= z -8e-28)
      (/ (* z x_m) (- z t))
      (if (<= z 2.15e-29) (* (/ x_m t) (- y z)) (* (/ x_m (- z t)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -8e-28) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 2.15e-29) {
		tmp = (x_m / t) * (y - z);
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9d+173)) then
        tmp = x_m / 1.0d0
    else if (z <= (-8d-28)) then
        tmp = (z * x_m) / (z - t)
    else if (z <= 2.15d-29) then
        tmp = (x_m / t) * (y - z)
    else
        tmp = (x_m / (z - t)) * z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -8e-28) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 2.15e-29) {
		tmp = (x_m / t) * (y - z);
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9e+173:
		tmp = x_m / 1.0
	elif z <= -8e-28:
		tmp = (z * x_m) / (z - t)
	elif z <= 2.15e-29:
		tmp = (x_m / t) * (y - z)
	else:
		tmp = (x_m / (z - t)) * z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9e+173)
		tmp = Float64(x_m / 1.0);
	elseif (z <= -8e-28)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	elseif (z <= 2.15e-29)
		tmp = Float64(Float64(x_m / t) * Float64(y - z));
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9e+173)
		tmp = x_m / 1.0;
	elseif (z <= -8e-28)
		tmp = (z * x_m) / (z - t);
	elseif (z <= 2.15e-29)
		tmp = (x_m / t) * (y - z);
	else
		tmp = (x_m / (z - t)) * z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9e+173], N[(x$95$m / 1.0), $MachinePrecision], If[LessEqual[z, -8e-28], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e-29], N[(N[(x$95$m / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -9.0000000000000004e173

   1. Initial program 56.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 88.8%

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

   if -9.0000000000000004e173 < z < -7.99999999999999977e-28

   1. Initial program 92.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 70.9

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

   7. Applied rewrites 70.9%

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

   if -7.99999999999999977e-28 < z < 2.1499999999999999e-29

   1. Initial program 94.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 78.9%

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

   if 2.1499999999999999e-29 < z

   1. Initial program 77.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 61.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-28}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{t} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{elif}\;z \leq 2.16 \cdot 10^{-32}:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot z\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9e+173)
    (/ x_m 1.0)
    (if (<= z -5.1e-89)
      (/ (* z x_m) (- z t))
      (if (<= z 2.16e-32) (* (/ x_m t) y) (* (/ x_m (- z t)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -5.1e-89) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 2.16e-32) {
		tmp = (x_m / t) * y;
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9d+173)) then
        tmp = x_m / 1.0d0
    else if (z <= (-5.1d-89)) then
        tmp = (z * x_m) / (z - t)
    else if (z <= 2.16d-32) then
        tmp = (x_m / t) * y
    else
        tmp = (x_m / (z - t)) * z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9e+173) {
		tmp = x_m / 1.0;
	} else if (z <= -5.1e-89) {
		tmp = (z * x_m) / (z - t);
	} else if (z <= 2.16e-32) {
		tmp = (x_m / t) * y;
	} else {
		tmp = (x_m / (z - t)) * z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9e+173:
		tmp = x_m / 1.0
	elif z <= -5.1e-89:
		tmp = (z * x_m) / (z - t)
	elif z <= 2.16e-32:
		tmp = (x_m / t) * y
	else:
		tmp = (x_m / (z - t)) * z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9e+173)
		tmp = Float64(x_m / 1.0);
	elseif (z <= -5.1e-89)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	elseif (z <= 2.16e-32)
		tmp = Float64(Float64(x_m / t) * y);
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9e+173)
		tmp = x_m / 1.0;
	elseif (z <= -5.1e-89)
		tmp = (z * x_m) / (z - t);
	elseif (z <= 2.16e-32)
		tmp = (x_m / t) * y;
	else
		tmp = (x_m / (z - t)) * z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9e+173], N[(x$95$m / 1.0), $MachinePrecision], If[LessEqual[z, -5.1e-89], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.16e-32], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -9.0000000000000004e173

   1. Initial program 56.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 88.8%

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

   if -9.0000000000000004e173 < z < -5.10000000000000004e-89

   1. Initial program 94.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 67.7

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

   7. Applied rewrites 67.7%

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

   if -5.10000000000000004e-89 < z < 2.1600000000000001e-32

   1. Initial program 93.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 75.0%

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

   if 2.1600000000000001e-32 < z

   1. Initial program 77.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 61.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq 2.16 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+111}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+209}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{y - t}{-z}, x\_m\right)\\ \end{array} \end{array} \]

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 -3.6e+111)
    (* (/ z (- z t)) x_m)
    (if (<= z 1.7e+209)
      (* (/ x_m (- t z)) (- y z))
      (fma x_m (/ (- y 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) {
	double tmp;
	if (z <= -3.6e+111) {
		tmp = (z / (z - t)) * x_m;
	} else if (z <= 1.7e+209) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = fma(x_m, ((y - t) / -z), 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 <= -3.6e+111)
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	elseif (z <= 1.7e+209)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = fma(x_m, Float64(Float64(y - t) / Float64(-z)), 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, -3.6e+111], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 1.7e+209], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y - t), $MachinePrecision] / (-z)), $MachinePrecision] + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.6000000000000002e111

   1. Initial program 64.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if -3.6000000000000002e111 < z < 1.6999999999999998e209

   1. Initial program 91.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 94.5

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

   4. Applied rewrites 94.5%

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

   if 1.6999999999999998e209 < z

   1. Initial program 72.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out-- N/A

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

      14. mul-1-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 88.3%

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

Alternative 8: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-32} \lor \neg \left(z \leq 3.8 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \end{array} \end{array} \]

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 (or (<= z -1.2e-32) (not (<= z 3.8e-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 ((z <= -1.2e-32) || !(z <= 3.8e-28)) {
		tmp = (z / (z - t)) * x_m;
	} else {
		tmp = (x_m / (t - z)) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.2d-32)) .or. (.not. (z <= 3.8d-28))) then
        tmp = (z / (z - t)) * x_m
    else
        tmp = (x_m / (t - z)) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.2e-32) || !(z <= 3.8e-28)) {
		tmp = (z / (z - t)) * x_m;
	} else {
		tmp = (x_m / (t - z)) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -1.2e-32) or not (z <= 3.8e-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 ((z <= -1.2e-32) || !(z <= 3.8e-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

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 <= -1.2e-32) || ~((z <= 3.8e-28)))
		tmp = (z / (z - t)) * x_m;
	else
		tmp = (x_m / (t - z)) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -1.2e-32], N[Not[LessEqual[z, 3.8e-28]], $MachinePrecision]], 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 2 regimes

2. if z < -1.2000000000000001e-32 or 3.80000000000000009e-28 < z

   1. Initial program 78.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.2000000000000001e-32 < z < 3.80000000000000009e-28

   1. Initial program 93.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-32} \lor \neg \left(z \leq 3.8 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-28} \lor \neg \left(z \leq 1.62 \cdot 10^{+75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, t, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \end{array} \end{array} \]

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 (or (<= z -8e-28) (not (<= z 1.62e+75)))
    (fma (/ x_m z) t x_m)
    (* (/ y 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 <= -8e-28) || !(z <= 1.62e+75)) {
		tmp = fma((x_m / z), t, x_m);
	} else {
		tmp = (y / 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 <= -8e-28) || !(z <= 1.62e+75))
		tmp = fma(Float64(x_m / z), t, x_m);
	else
		tmp = Float64(Float64(y / 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[Or[LessEqual[z, -8e-28], N[Not[LessEqual[z, 1.62e+75]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * t + x$95$m), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999977e-28 or 1.6200000000000001e75 < z

   1. Initial program 75.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 62.0

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

   7. Applied rewrites 62.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 62.6%

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

   if -7.99999999999999977e-28 < z < 1.6200000000000001e75

   1. Initial program 93.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-28} \lor \neg \left(z \leq 1.62 \cdot 10^{+75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-28} \lor \neg \left(z \leq 1.62 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \end{array} \end{array} \]

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 (or (<= z -8e-28) (not (<= z 1.62e+75))) (/ x_m 1.0) (* (/ y 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 <= -8e-28) || !(z <= 1.62e+75)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (y / 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 ((z <= (-8d-28)) .or. (.not. (z <= 1.62d+75))) then
        tmp = x_m / 1.0d0
    else
        tmp = (y / 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 ((z <= -8e-28) || !(z <= 1.62e+75)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (y / 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 (z <= -8e-28) or not (z <= 1.62e+75):
		tmp = x_m / 1.0
	else:
		tmp = (y / 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 <= -8e-28) || !(z <= 1.62e+75))
		tmp = Float64(x_m / 1.0);
	else
		tmp = Float64(Float64(y / 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 ((z <= -8e-28) || ~((z <= 1.62e+75)))
		tmp = x_m / 1.0;
	else
		tmp = (y / 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[Or[LessEqual[z, -8e-28], N[Not[LessEqual[z, 1.62e+75]], $MachinePrecision]], N[(x$95$m / 1.0), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999977e-28 or 1.6200000000000001e75 < z

   1. Initial program 75.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 62.4%

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

   if -7.99999999999999977e-28 < z < 1.6200000000000001e75

   1. Initial program 93.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-28} \lor \neg \left(z \leq 1.62 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-31} \lor \neg \left(z \leq 1.6 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \end{array} \end{array} \]

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 (or (<= z -4e-31) (not (<= z 1.6e+75))) (/ x_m 1.0) (* (/ x_m t) 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 ((z <= -4e-31) || !(z <= 1.6e+75)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (x_m / t) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-4d-31)) .or. (.not. (z <= 1.6d+75))) then
        tmp = x_m / 1.0d0
    else
        tmp = (x_m / t) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -4e-31) || !(z <= 1.6e+75)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (x_m / t) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -4e-31) or not (z <= 1.6e+75):
		tmp = x_m / 1.0
	else:
		tmp = (x_m / t) * 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 ((z <= -4e-31) || !(z <= 1.6e+75))
		tmp = Float64(x_m / 1.0);
	else
		tmp = Float64(Float64(x_m / t) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -4e-31) || ~((z <= 1.6e+75)))
		tmp = x_m / 1.0;
	else
		tmp = (x_m / t) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -4e-31], N[Not[LessEqual[z, 1.6e+75]], $MachinePrecision]], N[(x$95$m / 1.0), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4e-31 or 1.59999999999999992e75 < z

   1. Initial program 75.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 61.8%

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

   if -4e-31 < z < 1.59999999999999992e75

   1. Initial program 93.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 65.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-31} \lor \neg \left(z \leq 1.6 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

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 2e+17) (/ (* x_m (- y z)) (- t z)) (* (/ x_m (- t z)) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+17) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2d+17) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (x_m / (t - z)) * (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+17) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2e+17:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (t - z)) * (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e+17)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2e+17)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (x_m / (t - z)) * (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+17], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e17

   1. Initial program 90.5%

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

   if 2e17 < x

   1. Initial program 70.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. *-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 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 13: 97.1% 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 86.2%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
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 94.9

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

4. Applied rewrites 94.9%

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

Alternative 14: 35.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
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. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 95.5

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 30.7%

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

Developer Target 1: 97.0% 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 2024318 
(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)))