Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.5% → 97.6%

Time: 11.2s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} - \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{y - b}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, \frac{x}{t\_2} \cdot y\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-303}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (* (/ (- a t) z) (/ y (pow (- b y) 2.0)))
          (/ (fma (/ x z) y (- t a)) (- y b))))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_4 (fma z (/ (- t a) t_2) (* (/ x t_2) y))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -5e-303)
       t_3
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 4e+295) t_3 (if (<= t_3 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((a - t) / z) * (y / pow((b - y), 2.0))) - (fma((x / z), y, (t - a)) / (y - b));
	double t_2 = fma((b - y), z, y);
	double t_3 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_4 = fma(z, ((t - a) / t_2), ((x / t_2) * y));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -5e-303) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 4e+295) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(a - t) / z) * Float64(y / (Float64(b - y) ^ 2.0))) - Float64(fma(Float64(x / z), y, Float64(t - a)) / Float64(y - b)))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_4 = fma(z, Float64(Float64(t - a) / t_2), Float64(Float64(x / t_2) * y))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -5e-303)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 4e+295)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision] * N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x / z), $MachinePrecision] * y + N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x / t$95$2), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -5e-303], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 4e+295], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 3.9999999999999999e295 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 41.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 97.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 97.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999998e-303 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 3.9999999999999999e295

   1. Initial program 99.4%

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

   if -4.9999999999999998e-303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 13.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{t}{b - y} + \frac{x \cdot y}{z \cdot \left(b - y\right)}\right) - \left(\frac{a}{b - y} + \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}}{b - y} - \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{z}}, y, t - a\right)}{b - y} - \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{z}, y, \color{blue}{t - a}\right)}{b - y} - \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{\color{blue}{b - y}} - \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y \cdot \left(t - a\right)}{\color{blue}{{\left(b - y\right)}^{2} \cdot z}} \]

   5. Applied rewrites 85.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\right)\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{a - t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} - \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{y - b}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} - \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{t\_1}, \frac{x}{t\_1} \cdot y\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_3 (fma z (/ (- t a) t_1) (* (/ x t_1) y))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 4e+295) t_2 (if (<= t_2 INFINITY) t_3 (/ (- a t) (- y b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_3 = fma(z, ((t - a) / t_1), ((x / t_1) * y));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 4e+295) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (a - t) / (y - b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_3 = fma(z, Float64(Float64(t - a) / t_1), Float64(Float64(x / t_1) * y))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 4e+295)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(a - t) / Float64(y - b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(x / t$95$1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 4e+295], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 3.9999999999999999e295 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 41.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 97.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 97.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 3.9999999999999999e295

   1. Initial program 92.7%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 72.7

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\right)\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ t_2 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ z (fma (- b y) z y)) (- t a))) (t_2 (/ (- a t) (- y b))))
   (if (<= z -7.4e+25)
     t_2
     (if (<= z -2.4e-97)
       t_1
       (if (<= z 7.5e-154)
         (fma (/ (- a) y) z x)
         (if (<= z 6.2e+20) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z / fma((b - y), z, y)) * (t - a);
	double t_2 = (a - t) / (y - b);
	double tmp;
	if (z <= -7.4e+25) {
		tmp = t_2;
	} else if (z <= -2.4e-97) {
		tmp = t_1;
	} else if (z <= 7.5e-154) {
		tmp = fma((-a / y), z, x);
	} else if (z <= 6.2e+20) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z / fma(Float64(b - y), z, y)) * Float64(t - a))
	t_2 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -7.4e+25)
		tmp = t_2;
	elseif (z <= -2.4e-97)
		tmp = t_1;
	elseif (z <= 7.5e-154)
		tmp = fma(Float64(Float64(-a) / y), z, x);
	elseif (z <= 6.2e+20)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.4e+25], t$95$2, If[LessEqual[z, -2.4e-97], t$95$1, If[LessEqual[z, 7.5e-154], N[(N[((-a) / y), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 6.2e+20], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.3999999999999998e25 or 6.2e20 < z

   1. Initial program 49.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 83.4

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -7.3999999999999998e25 < z < -2.4e-97 or 7.5e-154 < z < 6.2e20

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if -2.4e-97 < z < 7.5e-154

   1. Initial program 89.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 68.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -6.6e+21)
     t_1
     (if (<= z 2.05e+18)
       (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -6.6e+21) {
		tmp = t_1;
	} else if (z <= 2.05e+18) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - t) / (y - b)
    if (z <= (-6.6d+21)) then
        tmp = t_1
    else if (z <= 2.05d+18) then
        tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -6.6e+21) {
		tmp = t_1;
	} else if (z <= 2.05e+18) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / (y - b)
	tmp = 0
	if z <= -6.6e+21:
		tmp = t_1
	elif z <= 2.05e+18:
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -6.6e+21)
		tmp = t_1;
	elseif (z <= 2.05e+18)
		tmp = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -6.6e+21)
		tmp = t_1;
	elseif (z <= 2.05e+18)
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+21], t$95$1, If[LessEqual[z, 2.05e+18], N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e21 or 2.05e18 < z

   1. Initial program 49.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 83.1

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -6.6e21 < z < 2.05e18

   1. Initial program 91.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -4.2e-31)
     t_1
     (if (<= z 9.5e-26) (/ (fma t z (* y x)) (+ (* (- b y) z) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -4.2e-31) {
		tmp = t_1;
	} else if (z <= 9.5e-26) {
		tmp = fma(t, z, (y * x)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -4.2e-31)
		tmp = t_1;
	elseif (z <= 9.5e-26)
		tmp = Float64(fma(t, z, Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-31], t$95$1, If[LessEqual[z, 9.5e-26], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999982e-31 or 9.4999999999999995e-26 < z

   1. Initial program 54.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 80.1

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -4.19999999999999982e-31 < z < 9.4999999999999995e-26

   1. Initial program 91.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -4.2e-31)
     t_1
     (if (<= z 9.5e-26) (/ (fma t z (* y x)) (fma (- b y) z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -4.2e-31) {
		tmp = t_1;
	} else if (z <= 9.5e-26) {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -4.2e-31)
		tmp = t_1;
	elseif (z <= 9.5e-26)
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-31], t$95$1, If[LessEqual[z, 9.5e-26], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999982e-31 or 9.4999999999999995e-26 < z

   1. Initial program 54.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 80.1

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -4.19999999999999982e-31 < z < 9.4999999999999995e-26

   1. Initial program 91.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - \mathsf{fma}\left(b, x, a\right)}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -3.6e-97)
     t_1
     (if (<= z 6e-27) (fma (/ (- t (fma b x a)) y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 6e-27) {
		tmp = fma(((t - fma(b, x, a)) / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 6e-27)
		tmp = fma(Float64(Float64(t - fma(b, x, a)) / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-97], t$95$1, If[LessEqual[z, 6e-27], N[(N[(N[(t - N[(b * x + a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999997e-97 or 6.0000000000000002e-27 < z

   1. Initial program 59.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 76.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999997e-97 < z < 6.0000000000000002e-27

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{t - \left(a + b \cdot x\right)}{y}, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - \mathsf{fma}\left(b, x, a\right)}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -3.6e-97)
     t_1
     (if (<= z 2.75e-33) (fma x z x) (if (<= z 4.2e+30) (/ (- a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 2.75e-33) {
		tmp = fma(x, z, x);
	} else if (z <= 4.2e+30) {
		tmp = -a / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 2.75e-33)
		tmp = fma(x, z, x);
	elseif (z <= 4.2e+30)
		tmp = Float64(Float64(-a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-97], t$95$1, If[LessEqual[z, 2.75e-33], N[(x * z + x), $MachinePrecision], If[LessEqual[z, 4.2e+30], N[((-a) / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.59999999999999997e-97 or 4.2e30 < z

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 40.1

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{t}{\color{blue}{b - y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.5%

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

   if -3.59999999999999997e-97 < z < 2.75e-33

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 49.9%

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

   if 2.75e-33 < z < 4.2e30

   1. Initial program 78.8%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \frac{-a}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+32}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e-97)
   (/ t b)
   (if (<= z 2.75e-33) (fma x z x) (if (<= z 1.55e+32) (/ (- a) b) (/ t b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t / b;
	} else if (z <= 2.75e-33) {
		tmp = fma(x, z, x);
	} else if (z <= 1.55e+32) {
		tmp = -a / b;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = Float64(t / b);
	elseif (z <= 2.75e-33)
		tmp = fma(x, z, x);
	elseif (z <= 1.55e+32)
		tmp = Float64(Float64(-a) / b);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.6e-97], N[(t / b), $MachinePrecision], If[LessEqual[z, 2.75e-33], N[(x * z + x), $MachinePrecision], If[LessEqual[z, 1.55e+32], N[((-a) / b), $MachinePrecision], N[(t / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.59999999999999997e-97 or 1.54999999999999997e32 < z

   1. Initial program 57.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 39.2

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

   5. Applied rewrites 39.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto \frac{t}{\color{blue}{b}} \]

   if -3.59999999999999997e-97 < z < 2.75e-33

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 49.9%

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

   if 2.75e-33 < z < 1.54999999999999997e32

   1. Initial program 81.3%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.5%

      \[\leadsto \frac{-a}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -3.6e-97) t_1 (if (<= z 1.5e-27) (fma (/ (- a) y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 1.5e-27) {
		tmp = fma((-a / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 1.5e-27)
		tmp = fma(Float64(Float64(-a) / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-97], t$95$1, If[LessEqual[z, 1.5e-27], N[(N[((-a) / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999997e-97 or 1.5000000000000001e-27 < z

   1. Initial program 59.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 76.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999997e-97 < z < 1.5000000000000001e-27

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 62.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -3.6e-97) t_1 (if (<= z 2.9e-132) (fma (/ t y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 2.9e-132) {
		tmp = fma((t / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 2.9e-132)
		tmp = fma(Float64(t / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-97], t$95$1, If[LessEqual[z, 2.9e-132], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999997e-97 or 2.89999999999999983e-132 < z

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 72.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999997e-97 < z < 2.89999999999999983e-132

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.2%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -2.8e-97) t_1 (if (<= z 2.7e-150) (* 1.0 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -2.8e-97) {
		tmp = t_1;
	} else if (z <= 2.7e-150) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - t) / (y - b)
    if (z <= (-2.8d-97)) then
        tmp = t_1
    else if (z <= 2.7d-150) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -2.8e-97) {
		tmp = t_1;
	} else if (z <= 2.7e-150) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / (y - b)
	tmp = 0
	if z <= -2.8e-97:
		tmp = t_1
	elif z <= 2.7e-150:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -2.8e-97)
		tmp = t_1;
	elseif (z <= 2.7e-150)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -2.8e-97)
		tmp = t_1;
	elseif (z <= 2.7e-150)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-97], t$95$1, If[LessEqual[z, 2.7e-150], N[(1.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000002e-97 or 2.7000000000000001e-150 < z

   1. Initial program 64.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 71.0

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -2.8000000000000002e-97 < z < 2.7000000000000001e-150

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 89.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 56.4%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+92}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.95e+71) t_1 (if (<= y 1.06e+92) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.95e+71) {
		tmp = t_1;
	} else if (y <= 1.06e+92) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-1.95d+71)) then
        tmp = t_1
    else if (y <= 1.06d+92) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.95e+71) {
		tmp = t_1;
	} else if (y <= 1.06e+92) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.95e+71:
		tmp = t_1
	elif y <= 1.06e+92:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.95e+71)
		tmp = t_1;
	elseif (y <= 1.06e+92)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.95e+71)
		tmp = t_1;
	elseif (y <= 1.06e+92)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.95e+71], t$95$1, If[LessEqual[y, 1.06e+92], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9500000000000001e71 or 1.05999999999999999e92 < y

   1. Initial program 58.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if -1.9500000000000001e71 < y < 1.05999999999999999e92

   1. Initial program 79.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b}} \]

      2. lower--.f64 51.6

         \[\leadsto \frac{\color{blue}{t - a}}{b} \]

   5. Applied rewrites 51.6%

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

Alternative 14: 44.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -3.6e-97) t_1 (if (<= z 3.6e+21) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 3.6e+21) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-3.6d-97)) then
        tmp = t_1
    else if (z <= 3.6d+21) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t_1;
	} else if (z <= 3.6e+21) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -3.6e-97:
		tmp = t_1
	elif z <= 3.6e+21:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 3.6e+21)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -3.6e-97)
		tmp = t_1;
	elseif (z <= 3.6e+21)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-97], t$95$1, If[LessEqual[z, 3.6e+21], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999997e-97 or 3.6e21 < z

   1. Initial program 57.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{t}{\color{blue}{b - y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.1%

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

   if -3.59999999999999997e-97 < z < 3.6e21

   1. Initial program 89.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e-97) (/ t b) (if (<= z 2.1e+18) (fma x z x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.6e-97) {
		tmp = t / b;
	} else if (z <= 2.1e+18) {
		tmp = fma(x, z, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = Float64(t / b);
	elseif (z <= 2.1e+18)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.6e-97], N[(t / b), $MachinePrecision], If[LessEqual[z, 2.1e+18], N[(x * z + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999997e-97 or 2.1e18 < z

   1. Initial program 57.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.9%

      \[\leadsto \frac{t}{\color{blue}{b}} \]

   if -3.59999999999999997e-97 < z < 2.1e18

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 46.3%

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

Alternative 16: 25.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, z, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma x z x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(x, z, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(x, z, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * z + x), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 31.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 23.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
9. Add Preprocessing

Alternative 17: 25.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. frac-2neg N/A

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

   15. metadata-eval N/A

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

   16. remove-double-neg N/A

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

   17. lower-/.f64 71.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-fma.f64 71.8

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

4. Applied rewrites 71.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 58.9

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

7. Applied rewrites 58.9%

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

8. Taylor expanded in z around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 23.3%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Developer Target 1: 73.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (((z * t) + (y * x)) / (y + (z * (b - y)))) - (a / ((b - y) + (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))