Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.8% → 86.2%

Time: 15.0s

Alternatives: 16

Speedup: 1.0×

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: 66.8% 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: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - y\right) \cdot z + y\\ t_2 := \left(t - a\right) \cdot z\\ t_3 := \frac{\mathsf{fma}\left(y, x, t\_2\right)}{t\_1}\\ t_4 := \frac{t\_2 + y \cdot x}{t\_1}\\ t_5 := \frac{a - t}{y - b}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot z, \frac{1}{x}\right)}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- b y) z) y))
        (t_2 (* (- t a) z))
        (t_3 (/ (fma y x t_2) t_1))
        (t_4 (/ (+ t_2 (* y x)) t_1))
        (t_5 (/ (- a t) (- y b))))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -5e-267)
       t_3
       (if (<= t_4 0.0)
         (/ 1.0 (fma (- b y) (* (/ 1.0 (fma (- t a) z (* y x))) z) (/ 1.0 x)))
         (if (<= t_4 2e+304) t_3 t_5))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b - y) * z) + y;
	double t_2 = (t - a) * z;
	double t_3 = fma(y, x, t_2) / t_1;
	double t_4 = (t_2 + (y * x)) / t_1;
	double t_5 = (a - t) / (y - b);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -5e-267) {
		tmp = t_3;
	} else if (t_4 <= 0.0) {
		tmp = 1.0 / fma((b - y), ((1.0 / fma((t - a), z, (y * x))) * z), (1.0 / x));
	} else if (t_4 <= 2e+304) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b - y) * z) + y)
	t_2 = Float64(Float64(t - a) * z)
	t_3 = Float64(fma(y, x, t_2) / t_1)
	t_4 = Float64(Float64(t_2 + Float64(y * x)) / t_1)
	t_5 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -5e-267)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = Float64(1.0 / fma(Float64(b - y), Float64(Float64(1.0 / fma(Float64(t - a), z, Float64(y * x))) * z), Float64(1.0 / x)));
	elseif (t_4 <= 2e+304)
		tmp = t_3;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * x + t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -5e-267], t$95$3, If[LessEqual[t$95$4, 0.0], N[(1.0 / N[(N[(b - y), $MachinePrecision] * N[(N[(1.0 / N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+304], t$95$3, t$95$5]]]]]]]]]

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 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.3%

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

   3. Taylor expanded in z around 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 70.3

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

   5. Applied rewrites 70.3%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999999e-267 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      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

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

      4. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -4.9999999999999999e-267 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 20.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 20.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 96.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 96.4

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

   9. Applied rewrites 96.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b - y, z \cdot \frac{1}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}, \color{blue}{\frac{1}{x}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot z, \frac{1}{x}\right)}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_2 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{1 - z} - \frac{\mathsf{fma}\left(t - a, \frac{z}{z - 1}, \frac{\left(z \cdot x\right) \cdot b}{\left(1 - z\right) \cdot \left(1 - z\right)}\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{t\_2} \cdot z, \frac{y}{t\_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_2 (fma (- t a) z (* y x))))
   (if (<= t_1 -5e+303)
     (-
      (/ x (- 1.0 z))
      (/
       (fma (- t a) (/ z (- z 1.0)) (/ (* (* z x) b) (* (- 1.0 z) (- 1.0 z))))
       y))
     (if (<= t_1 2e+304)
       (/ 1.0 (fma (- b y) (* (/ 1.0 t_2) z) (/ y t_2)))
       (/ (- a t) (- y b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_2 = fma((t - a), z, (y * x));
	double tmp;
	if (t_1 <= -5e+303) {
		tmp = (x / (1.0 - z)) - (fma((t - a), (z / (z - 1.0)), (((z * x) * b) / ((1.0 - z) * (1.0 - z)))) / y);
	} else if (t_1 <= 2e+304) {
		tmp = 1.0 / fma((b - y), ((1.0 / t_2) * z), (y / t_2));
	} else {
		tmp = (a - t) / (y - b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_2 = fma(Float64(t - a), z, Float64(y * x))
	tmp = 0.0
	if (t_1 <= -5e+303)
		tmp = Float64(Float64(x / Float64(1.0 - z)) - Float64(fma(Float64(t - a), Float64(z / Float64(z - 1.0)), Float64(Float64(Float64(z * x) * b) / Float64(Float64(1.0 - z) * Float64(1.0 - z)))) / y));
	elseif (t_1 <= 2e+304)
		tmp = Float64(1.0 / fma(Float64(b - y), Float64(Float64(1.0 / t_2) * z), Float64(y / t_2)));
	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[(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$2 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+303], N[(N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t - a), $MachinePrecision] * N[(z / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * x), $MachinePrecision] * b), $MachinePrecision] / N[(N[(1.0 - z), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(1.0 / N[(N[(b - y), $MachinePrecision] * N[(N[(1.0 / t$95$2), $MachinePrecision] * z), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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)))) < -4.9999999999999997e303

   1. Initial program 16.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1} + -1 \cdot \frac{\frac{z \cdot \left(t - a\right)}{z - 1} - -1 \cdot \frac{b \cdot \left(x \cdot z\right)}{{\left(z - 1\right)}^{2}}}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 70.7%

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

   if -4.9999999999999997e303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 88.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 97.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

   if 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{1 - z} - \frac{\mathsf{fma}\left(t - a, \frac{z}{z - 1}, \frac{\left(z \cdot x\right) \cdot b}{\left(1 - z\right) \cdot \left(1 - z\right)}\right)}{y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot z, \frac{y}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ t_2 := \frac{a - t}{y - b}\\ t_3 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{t\_1} \cdot z, \frac{y}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- t a) z (* y x)))
        (t_2 (/ (- a t) (- y b)))
        (t_3 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))))
   (if (<= t_3 -5e+303)
     t_2
     (if (<= t_3 2e+304)
       (/ 1.0 (fma (- b y) (* (/ 1.0 t_1) z) (/ y t_1)))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x));
	double t_2 = (a - t) / (y - b);
	double t_3 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double tmp;
	if (t_3 <= -5e+303) {
		tmp = t_2;
	} else if (t_3 <= 2e+304) {
		tmp = 1.0 / fma((b - y), ((1.0 / t_1) * z), (y / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t - a), z, Float64(y * x))
	t_2 = Float64(Float64(a - t) / Float64(y - b))
	t_3 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	tmp = 0.0
	if (t_3 <= -5e+303)
		tmp = t_2;
	elseif (t_3 <= 2e+304)
		tmp = Float64(1.0 / fma(Float64(b - y), Float64(Float64(1.0 / t_1) * z), Float64(y / t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $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]}, If[LessEqual[t$95$3, -5e+303], t$95$2, If[LessEqual[t$95$3, 2e+304], N[(1.0 / N[(N[(b - y), $MachinePrecision] * N[(N[(1.0 / t$95$1), $MachinePrecision] * z), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999997e303 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 11.4%

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

   3. Taylor expanded in z around 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 70.7

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

   5. Applied rewrites 70.7%

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

   if -4.9999999999999997e303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 88.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 97.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 97.6

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

   6. Applied rewrites 97.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b - y, z \cdot \frac{1}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}, \frac{y}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, \frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot z, \frac{y}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{x}{\frac{b \cdot z}{y} + 1}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{y - z \cdot y}\\ \mathbf{elif}\;z \leq 580000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -4.8e-8)
     t_1
     (if (<= z -5.5e-221)
       (/ x (+ (/ (* b z) y) 1.0))
       (if (<= z 7.8e-178)
         (/ (fma t z (* y x)) (- y (* z y)))
         (if (<= z 580000000.0) (* (/ y (fma (- b y) z y)) 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 <= -4.8e-8) {
		tmp = t_1;
	} else if (z <= -5.5e-221) {
		tmp = x / (((b * z) / y) + 1.0);
	} else if (z <= 7.8e-178) {
		tmp = fma(t, z, (y * x)) / (y - (z * y));
	} else if (z <= 580000000.0) {
		tmp = (y / fma((b - y), z, y)) * 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 <= -4.8e-8)
		tmp = t_1;
	elseif (z <= -5.5e-221)
		tmp = Float64(x / Float64(Float64(Float64(b * z) / y) + 1.0));
	elseif (z <= 7.8e-178)
		tmp = Float64(fma(t, z, Float64(y * x)) / Float64(y - Float64(z * y)));
	elseif (z <= 580000000.0)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * 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, -4.8e-8], t$95$1, If[LessEqual[z, -5.5e-221], N[(x / N[(N[(N[(b * z), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-178], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 580000000.0], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.79999999999999997e-8 or 5.8e8 < z

   1. Initial program 44.3%

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

   3. Taylor expanded in z around 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -4.79999999999999997e-8 < z < -5.49999999999999966e-221

   1. Initial program 83.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 80.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 80.6

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

   6. Applied rewrites 80.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 62.9

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

   9. Applied rewrites 62.9%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{x}{1 + \frac{b \cdot z}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

       \[\leadsto \frac{x}{1 + \frac{b \cdot z}{y}} \]

   if -5.49999999999999966e-221 < z < 7.8000000000000005e-178

   1. Initial program 99.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 71.6%

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

   if 7.8000000000000005e-178 < z < 5.8e8

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{x}{\frac{b \cdot z}{y} + 1}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{y - z \cdot y}\\ \mathbf{elif}\;z \leq 580000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -1.6e+43)
     t_1
     (if (<= z 9.2e+45) (/ (fma y x (* (- t a) z)) (+ (* (- 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 <= -1.6e+43) {
		tmp = t_1;
	} else if (z <= 9.2e+45) {
		tmp = fma(y, x, ((t - a) * z)) / (((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 <= -1.6e+43)
		tmp = t_1;
	elseif (z <= 9.2e+45)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / 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, -1.6e+43], t$95$1, If[LessEqual[z, 9.2e+45], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $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 < -1.60000000000000007e43 or 9.20000000000000049e45 < z

   1. Initial program 34.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 84.4

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

   5. Applied rewrites 84.4%

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

   if -1.60000000000000007e43 < z < 9.20000000000000049e45

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      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

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

      4. lower-fma.f64 88.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -0.27:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -0.27)
     t_1
     (if (<= z 2.1e+14) (/ (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 <= -0.27) {
		tmp = t_1;
	} else if (z <= 2.1e+14) {
		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 <= -0.27)
		tmp = t_1;
	elseif (z <= 2.1e+14)
		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, -0.27], t$95$1, If[LessEqual[z, 2.1e+14], 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 < -0.27000000000000002 or 2.1e14 < z

   1. Initial program 42.3%

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

   3. Taylor expanded in z around 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 82.6

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

   5. Applied rewrites 82.6%

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

   if -0.27000000000000002 < z < 2.1e14

   1. Initial program 88.8%

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

   3. Taylor expanded in a around 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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.27:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -9 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 580000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -9e-7)
     t_1
     (if (<= z 580000000.0) (* (/ y (fma (- b y) z y)) 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 <= -9e-7) {
		tmp = t_1;
	} else if (z <= 580000000.0) {
		tmp = (y / fma((b - y), z, y)) * 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 <= -9e-7)
		tmp = t_1;
	elseif (z <= 580000000.0)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * 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, -9e-7], t$95$1, If[LessEqual[z, 580000000.0], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999959e-7 or 5.8e8 < z

   1. Initial program 44.3%

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

   3. Taylor expanded in z around 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -8.99999999999999959e-7 < z < 5.8e8

   1. Initial program 88.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 580000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+128}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 3.5:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.25e+128)
   (/ (- t a) (- y))
   (if (<= z -2.2e-10)
     (/ (- t a) b)
     (if (<= z 3.5) (/ x (- 1.0 z)) (/ a (- y b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.25e+128) {
		tmp = (t - a) / -y;
	} else if (z <= -2.2e-10) {
		tmp = (t - a) / b;
	} else if (z <= 3.5) {
		tmp = x / (1.0 - z);
	} else {
		tmp = a / (y - b);
	}
	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) :: tmp
    if (z <= (-1.25d+128)) then
        tmp = (t - a) / -y
    else if (z <= (-2.2d-10)) then
        tmp = (t - a) / b
    else if (z <= 3.5d0) then
        tmp = x / (1.0d0 - z)
    else
        tmp = a / (y - b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.25e+128) {
		tmp = (t - a) / -y;
	} else if (z <= -2.2e-10) {
		tmp = (t - a) / b;
	} else if (z <= 3.5) {
		tmp = x / (1.0 - z);
	} else {
		tmp = a / (y - b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.25e+128:
		tmp = (t - a) / -y
	elif z <= -2.2e-10:
		tmp = (t - a) / b
	elif z <= 3.5:
		tmp = x / (1.0 - z)
	else:
		tmp = a / (y - b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.25e+128)
		tmp = Float64(Float64(t - a) / Float64(-y));
	elseif (z <= -2.2e-10)
		tmp = Float64(Float64(t - a) / b);
	elseif (z <= 3.5)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(a / Float64(y - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.25e+128)
		tmp = (t - a) / -y;
	elseif (z <= -2.2e-10)
		tmp = (t - a) / b;
	elseif (z <= 3.5)
		tmp = x / (1.0 - z);
	else
		tmp = a / (y - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.25e+128], N[(N[(t - a), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[z, -2.2e-10], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 3.5], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.25e128

   1. Initial program 18.9%

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

      1. lift-+.f64 N/A

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

      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

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

      4. lower-fma.f64 19.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 19.0

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

   4. Applied rewrites 19.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   6. 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 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{t - a}{-1 \cdot \color{blue}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto \frac{t - a}{-y} \]

   if -1.25e128 < z < -2.1999999999999999e-10

   1. Initial program 76.3%

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

   3. Taylor expanded in 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 59.2

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

   5. Applied rewrites 59.2%

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

   if -2.1999999999999999e-10 < z < 3.5

   1. Initial program 88.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

   if 3.5 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 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

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+128}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 3.5:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.00078:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -3.7e-16) t_1 (if (<= z 0.00078) (/ x (- 1.0 z)) 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 <= -3.7e-16) {
		tmp = t_1;
	} else if (z <= 0.00078) {
		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 = (a - t) / (y - b)
    if (z <= (-3.7d-16)) then
        tmp = t_1
    else if (z <= 0.00078d0) 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 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.7e-16) {
		tmp = t_1;
	} else if (z <= 0.00078) {
		tmp = x / (1.0 - z);
	} 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 <= -3.7e-16:
		tmp = t_1
	elif z <= 0.00078:
		tmp = x / (1.0 - z)
	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 <= -3.7e-16)
		tmp = t_1;
	elseif (z <= 0.00078)
		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 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -3.7e-16)
		tmp = t_1;
	elseif (z <= 0.00078)
		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[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.7e-16], t$95$1, If[LessEqual[z, 0.00078], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e-16 or 7.79999999999999986e-4 < z

   1. Initial program 44.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.7e-16 < z < 7.79999999999999986e-4

   1. Initial program 88.8%

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

   3. Taylor expanded in 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 50.1

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

   5. Applied rewrites 50.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 0.00078:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -6900000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -6900000000.0) t_1 (if (<= y 1.95) (/ (- 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 <= -6900000000.0) {
		tmp = t_1;
	} else if (y <= 1.95) {
		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 <= (-6900000000.0d0)) then
        tmp = t_1
    else if (y <= 1.95d0) 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 <= -6900000000.0) {
		tmp = t_1;
	} else if (y <= 1.95) {
		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 <= -6900000000.0:
		tmp = t_1
	elif y <= 1.95:
		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 <= -6900000000.0)
		tmp = t_1;
	elseif (y <= 1.95)
		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 <= -6900000000.0)
		tmp = t_1;
	elseif (y <= 1.95)
		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, -6900000000.0], t$95$1, If[LessEqual[y, 1.95], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9e9 or 1.94999999999999996 < y

   1. Initial program 50.8%

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

   3. Taylor expanded in 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 48.3

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

   5. Applied rewrites 48.3%

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

   if -6.9e9 < y < 1.94999999999999996

   1. Initial program 79.7%

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

   3. Taylor expanded in 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 60.0

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

   5. Applied rewrites 60.0%

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

Alternative 11: 41.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.9e-16)
   (/ t (- b y))
   (if (<= z 1.1e+16) (/ x (- 1.0 z)) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.9e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.1e+16) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -a / b;
	}
	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) :: tmp
    if (z <= (-3.9d-16)) then
        tmp = t / (b - y)
    else if (z <= 1.1d+16) then
        tmp = x / (1.0d0 - z)
    else
        tmp = -a / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.9e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.1e+16) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -3.9e-16:
		tmp = t / (b - y)
	elif z <= 1.1e+16:
		tmp = x / (1.0 - z)
	else:
		tmp = -a / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.9e-16)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 1.1e+16)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -3.9e-16)
		tmp = t / (b - y);
	elseif (z <= 1.1e+16)
		tmp = x / (1.0 - z);
	else
		tmp = -a / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.9e-16], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e+16], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.89999999999999977e-16

   1. Initial program 40.1%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.7%

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

   if -3.89999999999999977e-16 < z < 1.1e16

   1. Initial program 88.3%

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

   3. Taylor expanded in 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 48.9

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

   5. Applied rewrites 48.9%

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

   if 1.1e16 < 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 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 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

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 39.8%

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

Alternative 12: 41.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.9e-16)
   (/ t (- b y))
   (if (<= z 1.05e+15) (fma x z x) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.9e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.05e+15) {
		tmp = fma(x, z, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.9e-16)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 1.05e+15)
		tmp = fma(x, z, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.9e-16], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+15], N[(x * z + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.89999999999999977e-16

   1. Initial program 40.1%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.7%

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

   if -3.89999999999999977e-16 < z < 1.05e15

   1. Initial program 88.3%

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

   3. Taylor expanded in 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 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.7%

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

   if 1.05e15 < 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 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 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

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 39.8%

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

Alternative 13: 37.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -9e-7) (/ t b) (if (<= z 1.05e+15) (fma x z x) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -9e-7) {
		tmp = t / b;
	} else if (z <= 1.05e+15) {
		tmp = fma(x, z, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -9e-7)
		tmp = Float64(t / b);
	elseif (z <= 1.05e+15)
		tmp = fma(x, z, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -9e-7], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.05e+15], N[(x * z + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.99999999999999959e-7

   1. Initial program 39.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 26.6%

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

   if -8.99999999999999959e-7 < z < 1.05e15

   1. Initial program 87.7%

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

   3. Taylor expanded in 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 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 47.9%

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

   if 1.05e15 < 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 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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 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

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 39.8%

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

Alternative 14: 35.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.1e-5) (* 1.0 x) (if (<= y 1.6e-75) (/ t b) (fma x z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.1e-5) {
		tmp = 1.0 * x;
	} else if (y <= 1.6e-75) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.1e-5)
		tmp = Float64(1.0 * x);
	elseif (y <= 1.6e-75)
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.1e-5], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 1.6e-75], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.09999999999999987e-5

   1. Initial program 56.5%

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

      1. lift-+.f64 N/A

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

      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

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

      4. lower-fma.f64 56.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 56.5

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 31.6%

       \[\leadsto x \cdot 1 \]

   if -6.09999999999999987e-5 < y < 1.59999999999999988e-75

   1. Initial program 78.4%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.2%

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

   if 1.59999999999999988e-75 < y

   1. Initial program 54.1%

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

   3. Taylor expanded in 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 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.2% 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 64.1%

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

   1. lift-+.f64 N/A

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

   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

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

   4. lower-fma.f64 64.1

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 32.4

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

7. Applied rewrites 32.4%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 23.4%

    \[\leadsto x \cdot 1 \]

11. Final simplification 23.4%

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

Alternative 16: 4.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return z * 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 = z * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in 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 31.5

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

5. Applied rewrites 31.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 23.3%

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

9. Taylor expanded in z around inf

   \[\leadsto x \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 3.2%

    \[\leadsto x \cdot z \]

12. Final simplification 3.2%

    \[\leadsto z \cdot x \]
13. Add Preprocessing

Developer Target 1: 73.4% 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 2024235 
(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)))))