Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.8% → 87.1%

Time: 13.5s

Alternatives: 17

Speedup: 1.3×

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: 87.1% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- y b) (- y b)))
        (t_2 (* (- b y) z))
        (t_3 (/ (- (* y x) (* (- a t) z)) (+ t_2 y)))
        (t_4 (/ (- a t) (- y b))))
   (if (<= t_3 (- INFINITY))
     (-
      (/ x (- 1.0 z))
      (/
       (fma z (/ (- a t) (- 1.0 z)) (/ (* (* z x) b) (* (- z 1.0) (- z 1.0))))
       y))
     (if (<= t_3 -1e-304)
       t_3
       (if (<= t_3 0.0)
         (-
          t_4
          (/
           (fma
            (/ (- t a) t_1)
            y
            (-
             (fma
              (/ y (- b y))
              x
              (* (/ y t_2) (fma (- y) (/ x (- b y)) (/ (* (- t a) y) t_1))))))
           z))
         (if (<= t_3 2e+304) t_3 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y - b) * (y - b);
	double t_2 = (b - y) * z;
	double t_3 = ((y * x) - ((a - t) * z)) / (t_2 + y);
	double t_4 = (a - t) / (y - b);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x / (1.0 - z)) - (fma(z, ((a - t) / (1.0 - z)), (((z * x) * b) / ((z - 1.0) * (z - 1.0)))) / y);
	} else if (t_3 <= -1e-304) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_4 - (fma(((t - a) / t_1), y, -fma((y / (b - y)), x, ((y / t_2) * fma(-y, (x / (b - y)), (((t - a) * y) / t_1))))) / z);
	} else if (t_3 <= 2e+304) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y - b) * Float64(y - b))
	t_2 = Float64(Float64(b - y) * z)
	t_3 = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(t_2 + y))
	t_4 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(1.0 - z)) - Float64(fma(z, Float64(Float64(a - t) / Float64(1.0 - z)), Float64(Float64(Float64(z * x) * b) / Float64(Float64(z - 1.0) * Float64(z - 1.0)))) / y));
	elseif (t_3 <= -1e-304)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(t_4 - Float64(fma(Float64(Float64(t - a) / t_1), y, Float64(-fma(Float64(y / Float64(b - y)), x, Float64(Float64(y / t_2) * fma(Float64(-y), Float64(x / Float64(b - y)), Float64(Float64(Float64(t - a) * y) / t_1)))))) / z));
	elseif (t_3 <= 2e+304)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 11.2

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

   4. Applied rewrites 11.2%

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

   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 57.3

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

   7. Applied rewrites 57.3%

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

   9. Applied rewrites 57.1%

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

   10. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-frac2 N/A

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

       6. mul-1-neg N/A

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-in N/A

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. lower-/.f64 N/A

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

       13. mul-1-neg N/A

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

       14. sub-neg N/A

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

       15. lower--.f64 N/A

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

       16. lower-/.f64 N/A

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

   12. Applied rewrites 68.8%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999971e-305 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

   if -9.99999999999999971e-305 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 17.8%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 93.1%

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

   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 4 regimes into one program.

4. Final simplification 90.7%

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

Alternative 2: 84.8% accurate, 0.8× speedup

Math

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

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)
       (/ (- (* y x) (* (- a t) 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 = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - t) / (y - b)
    if (z <= (-1.6d+43)) then
        tmp = t_1
    else if (z <= 9.2d+45) then
        tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -1.6e+43) {
		tmp = t_1;
	} else if (z <= 9.2e+45) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / (y - b)
	tmp = 0
	if z <= -1.6e+43:
		tmp = t_1
	elif z <= 9.2e+45:
		tmp = ((y * x) - ((a - t) * 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(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -1.6e+43)
		tmp = t_1;
	elseif (z <= 9.2e+45)
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+43], t$95$1, If[LessEqual[z, 9.2e+45], N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $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. Recombined 2 regimes into one program.

4. Final simplification 86.6%

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

Alternative 3: 73.8% accurate, 0.9× speedup

Math

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

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 y x (* t z)) (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(y, x, (t * z)) / 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(y, x, Float64(t * z)) / 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[(y * x + N[(t * z), $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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 79.6%

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

Alternative 4: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4400000:\\ \;\;\;\;\frac{1}{y} \cdot \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -7.5e-16)
     t_1
     (if (<= z 4400000.0) (* (/ 1.0 y) (fma (- t a) 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 <= -7.5e-16) {
		tmp = t_1;
	} else if (z <= 4400000.0) {
		tmp = (1.0 / y) * fma((t - a), 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 <= -7.5e-16)
		tmp = t_1;
	elseif (z <= 4400000.0)
		tmp = Float64(Float64(1.0 / y) * fma(Float64(t - a), z, Float64(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, -7.5e-16], t$95$1, If[LessEqual[z, 4400000.0], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e-16 or 4.4e6 < z

   1. Initial program 44.1%

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

   3. Taylor expanded in 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 81.4

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

   5. Applied rewrites 81.4%

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

   if -7.5e-16 < z < 4.4e6

   1. Initial program 89.0%

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

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. clear-num N/A

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

      16. flip-+ N/A

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

      17. lift-+.f64 N/A

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 61.9

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

   7. Applied rewrites 61.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4400000:\\ \;\;\;\;\frac{1}{y} \cdot \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 820000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -9.2e-7)
     t_1
     (if (<= z 820000000.0) (* (/ 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 <= -9.2e-7) {
		tmp = t_1;
	} else if (z <= 820000000.0) {
		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 <= -9.2e-7)
		tmp = t_1;
	elseif (z <= 820000000.0)
		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, -9.2e-7], t$95$1, If[LessEqual[z, 820000000.0], 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 < -9.1999999999999998e-7 or 8.2e8 < 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 -9.1999999999999998e-7 < z < 8.2e8

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 820000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+128}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 1800000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.25e+128)
   (/ (- t a) (- y))
   (if (<= z -1.8e-7)
     (/ (- t a) b)
     (if (<= z 1800000.0) (/ 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 <= -1.8e-7) {
		tmp = (t - a) / b;
	} else if (z <= 1800000.0) {
		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 <= (-1.8d-7)) then
        tmp = (t - a) / b
    else if (z <= 1800000.0d0) 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 <= -1.8e-7) {
		tmp = (t - a) / b;
	} else if (z <= 1800000.0) {
		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 <= -1.8e-7:
		tmp = (t - a) / b
	elif z <= 1800000.0:
		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 <= -1.8e-7)
		tmp = Float64(Float64(t - a) / b);
	elseif (z <= 1800000.0)
		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 <= -1.8e-7)
		tmp = (t - a) / b;
	elseif (z <= 1800000.0)
		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, -1.8e-7], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 1800000.0], 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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 14.7

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

   4. Applied rewrites 14.7%

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

   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 < -1.79999999999999997e-7

   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 -1.79999999999999997e-7 < z < 1.8e6

   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 1.8e6 < 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+128}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 1800000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+171}:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, z, x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+171)
   (/ a y)
   (if (<= z -9.5e-7)
     (/ t b)
     (if (<= z 1.05e+15) (fma (fma x z x) z x) (/ (- a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e+171) {
		tmp = a / y;
	} else if (z <= -9.5e-7) {
		tmp = t / b;
	} else if (z <= 1.05e+15) {
		tmp = fma(fma(x, z, x), z, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e+171)
		tmp = Float64(a / y);
	elseif (z <= -9.5e-7)
		tmp = Float64(t / b);
	elseif (z <= 1.05e+15)
		tmp = fma(fma(x, z, x), z, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e+171], N[(a / y), $MachinePrecision], If[LessEqual[z, -9.5e-7], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.05e+15], N[(N[(x * z + x), $MachinePrecision] * z + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.84999999999999999e171

   1. Initial program 16.1%

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

   3. Taylor expanded in a around 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 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.2%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{a}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.3%

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

   if -1.84999999999999999e171 < z < -9.5000000000000001e-7

   1. Initial program 60.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.0%

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

   if -9.5000000000000001e-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

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

   8. Applied rewrites 48.2%

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

   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 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+171}:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, z, x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+171)
   (/ a y)
   (if (<= z -9.5e-7)
     (/ t b)
     (if (<= z 4.4e+14) (fma (fma x z x) z x) (/ a y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e+171) {
		tmp = a / y;
	} else if (z <= -9.5e-7) {
		tmp = t / b;
	} else if (z <= 4.4e+14) {
		tmp = fma(fma(x, z, x), z, x);
	} else {
		tmp = a / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e+171)
		tmp = Float64(a / y);
	elseif (z <= -9.5e-7)
		tmp = Float64(t / b);
	elseif (z <= 4.4e+14)
		tmp = fma(fma(x, z, x), z, x);
	else
		tmp = Float64(a / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e+171], N[(a / y), $MachinePrecision], If[LessEqual[z, -9.5e-7], N[(t / b), $MachinePrecision], If[LessEqual[z, 4.4e+14], N[(N[(x * z + x), $MachinePrecision] * z + x), $MachinePrecision], N[(a / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999999e171 or 4.4e14 < z

   1. Initial program 37.1%

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

   3. Taylor expanded in a around 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 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.6%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{a}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.5%

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

   if -1.84999999999999999e171 < z < -9.5000000000000001e-7

   1. Initial program 60.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.0%

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

   if -9.5000000000000001e-7 < z < 4.4e14

   1. Initial program 88.4%

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

   3. Taylor expanded in 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 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.5%

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

Alternative 9: 35.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+171}:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+171)
   (/ a y)
   (if (<= z -9e-7) (/ t b) (if (<= z 4.4e+14) (fma x z x) (/ a y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e+171) {
		tmp = a / y;
	} else if (z <= -9e-7) {
		tmp = t / b;
	} else if (z <= 4.4e+14) {
		tmp = fma(x, z, x);
	} else {
		tmp = a / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e+171)
		tmp = Float64(a / y);
	elseif (z <= -9e-7)
		tmp = Float64(t / b);
	elseif (z <= 4.4e+14)
		tmp = fma(x, z, x);
	else
		tmp = Float64(a / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e+171], N[(a / y), $MachinePrecision], If[LessEqual[z, -9e-7], N[(t / b), $MachinePrecision], If[LessEqual[z, 4.4e+14], N[(x * z + x), $MachinePrecision], N[(a / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999999e171 or 4.4e14 < z

   1. Initial program 37.1%

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

   3. Taylor expanded in a around 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 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.6%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{a}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.5%

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

   if -1.84999999999999999e171 < z < -8.99999999999999959e-7

   1. Initial program 60.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.0%

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

   if -8.99999999999999959e-7 < z < 4.4e14

   1. Initial program 88.4%

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

   3. Taylor expanded in 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 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.3%

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

Alternative 10: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -2.4e-8) t_1 (if (<= z 4400000.0) (fma (/ t y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -2.4e-8) {
		tmp = t_1;
	} else if (z <= 4400000.0) {
		tmp = fma((t / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -2.4e-8)
		tmp = t_1;
	elseif (z <= 4400000.0)
		tmp = fma(Float64(t / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e-8], t$95$1, If[LessEqual[z, 4400000.0], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999998e-8 or 4.4e6 < 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 -2.39999999999999998e-8 < z < 4.4e6

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 52.3

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 56.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 2.5e-5) (/ 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 <= 2.5e-5) {
		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 <= 2.5d-5) 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 <= 2.5e-5) {
		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 <= 2.5e-5:
		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 <= 2.5e-5)
		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 <= 2.5e-5)
		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, 2.5e-5], 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 2.50000000000000012e-5 < 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 < 2.50000000000000012e-5

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -620000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -620000000.0) t_1 (if (<= y 1.75) (/ (- 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 <= -620000000.0) {
		tmp = t_1;
	} else if (y <= 1.75) {
		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 <= (-620000000.0d0)) then
        tmp = t_1
    else if (y <= 1.75d0) 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 <= -620000000.0) {
		tmp = t_1;
	} else if (y <= 1.75) {
		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 <= -620000000.0:
		tmp = t_1
	elif y <= 1.75:
		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 <= -620000000.0)
		tmp = t_1;
	elseif (y <= 1.75)
		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 <= -620000000.0)
		tmp = t_1;
	elseif (y <= 1.75)
		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, -620000000.0], t$95$1, If[LessEqual[y, 1.75], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e8 or 1.75 < 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 -6.2e8 < y < 1.75

   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 13: 41.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e-16)
   (/ t (- b y))
   (if (<= z 1.05e+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 <= -3.8e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.05e+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 <= (-3.8d-16)) then
        tmp = t / (b - y)
    else if (z <= 1.05d+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 <= -3.8e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.05e+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 <= -3.8e-16:
		tmp = t / (b - y)
	elif z <= 1.05e+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 <= -3.8e-16)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 1.05e+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 <= -3.8e-16)
		tmp = t / (b - y);
	elseif (z <= 1.05e+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, -3.8e-16], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+16], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.80000000000000012e-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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 32.2

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

   5. Applied rewrites 32.2%

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

   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 -3.80000000000000012e-16 < z < 1.05e16

   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 1.05e16 < 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: 41.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, z, x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e-16)
   (/ t (- b y))
   (if (<= z 1.05e+15) (fma (fma x z x) z x) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.8e-16) {
		tmp = t / (b - y);
	} else if (z <= 1.05e+15) {
		tmp = fma(fma(x, z, x), z, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.8e-16)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 1.05e+15)
		tmp = fma(fma(x, z, x), z, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.8e-16], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+15], N[(N[(x * z + x), $MachinePrecision] * z + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.80000000000000012e-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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 32.2

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

   5. Applied rewrites 32.2%

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

   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 -3.80000000000000012e-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

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

   8. Applied rewrites 48.9%

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

   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 15: 34.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0033:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.0033) (/ a y) (if (<= z 4.4e+14) (fma x z x) (/ a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.0033) {
		tmp = a / y;
	} else if (z <= 4.4e+14) {
		tmp = fma(x, z, x);
	} else {
		tmp = a / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.0033)
		tmp = Float64(a / y);
	elseif (z <= 4.4e+14)
		tmp = fma(x, z, x);
	else
		tmp = Float64(a / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.0033], N[(a / y), $MachinePrecision], If[LessEqual[z, 4.4e+14], N[(x * z + x), $MachinePrecision], N[(a / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0033 or 4.4e14 < z

   1. Initial program 43.1%

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

   3. Taylor expanded in a around 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 32.6

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

   5. Applied rewrites 32.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.3%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{a}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.5%

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

   if -0.0033 < z < 4.4e14

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 47.6%

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

Alternative 16: 26.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing

Alternative 17: 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)))))