Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 65.6% → 87.1%

Time: 14.1s

Alternatives: 11

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: 65.6% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{z \cdot \left(t - a\right) + x \cdot y}{t\_1}\\ t_4 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot t\_4}, \frac{y}{t\_4}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y - z \cdot a}{t\_4}, 1, \frac{z \cdot t}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* z (- t a)) (* x y)) t_1))
        (t_4 (fma z (- b y) y)))
   (if (<= t_3 (- INFINITY))
     (* x (fma z (/ (- t a) (* x t_4)) (/ y t_4)))
     (if (<= t_3 -5e-222)
       (/ (fma (- t a) z (* x y)) t_1)
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 1e+255)
           (fma (/ (- (* x y) (* z a)) t_4) 1.0 (/ (* z t) t_4))
           t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = ((z * (t - a)) + (x * y)) / t_1;
	double t_4 = fma(z, (b - y), y);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = x * fma(z, ((t - a) / (x * t_4)), (y / t_4));
	} else if (t_3 <= -5e-222) {
		tmp = fma((t - a), z, (x * y)) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+255) {
		tmp = fma((((x * y) - (z * a)) / t_4), 1.0, ((z * t) / t_4));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / t_1)
	t_4 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(x * fma(z, Float64(Float64(t - a) / Float64(x * t_4)), Float64(y / t_4)));
	elseif (t_3 <= -5e-222)
		tmp = Float64(fma(Float64(t - a), z, Float64(x * y)) / t_1);
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+255)
		tmp = fma(Float64(Float64(Float64(x * y) - Float64(z * a)) / t_4), 1.0, Float64(Float64(z * t) / t_4));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(x * N[(z * N[(N[(t - a), $MachinePrecision] / N[(x * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-222], N[(N[(N[(t - a), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+255], N[(N[(N[(N[(x * y), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0 + N[(N[(z * t), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

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 26.5%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. --lowering--.f64 79.5

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

   5. Simplified 79.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.00000000000000008e-222

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   if -5.00000000000000008e-222 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or 9.99999999999999988e254 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 20.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 80.5

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

   5. Simplified 80.5%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.99999999999999988e254

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

   5. Simplified 88.6%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y \cdot x - z \cdot a}{\mathsf{fma}\left(z, b - y, y\right)}, 1, \frac{t \cdot z}{\mathsf{fma}\left(z, b - y, y\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot \mathsf{fma}\left(z, b - y, y\right)}, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y - z \cdot a}{\mathsf{fma}\left(z, b - y, y\right)}, 1, \frac{z \cdot t}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{z \cdot \left(t - a\right) + x \cdot y}{t\_1}\\ t_5 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot t\_5}, \frac{y}{t\_5}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{+255}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma (- t a) z (* x y)) t_1))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ (+ (* z (- t a)) (* x y)) t_1))
        (t_5 (fma z (- b y) y)))
   (if (<= t_4 (- INFINITY))
     (* x (fma z (/ (- t a) (* x t_5)) (/ y t_5)))
     (if (<= t_4 -5e-222)
       t_2
       (if (<= t_4 0.0) t_3 (if (<= t_4 1e+255) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = fma((t - a), z, (x * y)) / t_1;
	double t_3 = (t - a) / (b - y);
	double t_4 = ((z * (t - a)) + (x * y)) / t_1;
	double t_5 = fma(z, (b - y), y);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = x * fma(z, ((t - a) / (x * t_5)), (y / t_5));
	} else if (t_4 <= -5e-222) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_3;
	} else if (t_4 <= 1e+255) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(fma(Float64(t - a), z, Float64(x * y)) / t_1)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / t_1)
	t_5 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(x * fma(z, Float64(Float64(t - a) / Float64(x * t_5)), Float64(y / t_5)));
	elseif (t_4 <= -5e-222)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = t_3;
	elseif (t_4 <= 1e+255)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - a), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(x * N[(z * N[(N[(t - a), $MachinePrecision] / N[(x * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-222], t$95$2, If[LessEqual[t$95$4, 0.0], t$95$3, If[LessEqual[t$95$4, 1e+255], t$95$2, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.5%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. --lowering--.f64 79.5

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

   5. Simplified 79.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.00000000000000008e-222 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.99999999999999988e254

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   if -5.00000000000000008e-222 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or 9.99999999999999988e254 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 20.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot \mathsf{fma}\left(z, b - y, y\right)}, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8.8e+24)
     t_1
     (if (<= z 9e+14) (/ (fma (- t a) z (* x y)) (+ y (* z (- b y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -8.8e+24) {
		tmp = t_1;
	} else if (z <= 9e+14) {
		tmp = fma((t - a), z, (x * y)) / (y + (z * (b - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -8.8e+24)
		tmp = t_1;
	elseif (z <= 9e+14)
		tmp = Float64(fma(Float64(t - a), z, Float64(x * y)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.8e+24], t$95$1, If[LessEqual[z, 9e+14], N[(N[(N[(t - a), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.80000000000000007e24 or 9e14 < 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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 85.6

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

   5. Simplified 85.6%

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

   if -8.80000000000000007e24 < z < 9e14

   1. Initial program 86.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 86.5

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

   4. Applied egg-rr 86.5%

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

Alternative 4: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1e-5) t_1 (if (<= z 2.65e-12) (fma z (/ (- t a) y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e-5) {
		tmp = t_1;
	} else if (z <= 2.65e-12) {
		tmp = fma(z, ((t - a) / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e-5)
		tmp = t_1;
	elseif (z <= 2.65e-12)
		tmp = fma(z, Float64(Float64(t - a) / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-5], t$95$1, If[LessEqual[z, 2.65e-12], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000008e-5 or 2.64999999999999982e-12 < z

   1. Initial program 47.9%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 85.4

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

   5. Simplified 85.4%

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

   if -1.00000000000000008e-5 < z < 2.64999999999999982e-12

   1. Initial program 85.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 N/A

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

      15. /-lowering-/.f64 56.2

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. /-lowering-/.f64 N/A

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

      3. --lowering--.f64 61.3

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

   8. Simplified 61.3%

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

Alternative 5: 66.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(z, -\frac{a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5.5e-6) t_1 (if (<= z 5.1e-126) (fma z (- (/ a y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -5.5e-6) {
		tmp = t_1;
	} else if (z <= 5.1e-126) {
		tmp = fma(z, -(a / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -5.5e-6)
		tmp = t_1;
	elseif (z <= 5.1e-126)
		tmp = fma(z, Float64(-Float64(a / y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e-6], t$95$1, If[LessEqual[z, 5.1e-126], N[(z * (-N[(a / y), $MachinePrecision]) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999999e-6 or 5.10000000000000002e-126 < z

   1. Initial program 53.9%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 78.8

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

   5. Simplified 78.8%

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

   if -5.4999999999999999e-6 < z < 5.10000000000000002e-126

   1. Initial program 84.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 N/A

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

      15. /-lowering-/.f64 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 55.2

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

   8. Simplified 55.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(z, -\frac{a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.5e-5) t_1 (if (<= z 3.3e-16) (fma z (/ t y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.5e-5) {
		tmp = t_1;
	} else if (z <= 3.3e-16) {
		tmp = fma(z, (t / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.5e-5)
		tmp = t_1;
	elseif (z <= 3.3e-16)
		tmp = fma(z, Float64(t / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e-5], t$95$1, If[LessEqual[z, 3.3e-16], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000004e-5 or 3.29999999999999988e-16 < z

   1. Initial program 47.9%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 85.4

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

   5. Simplified 85.4%

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

   if -1.50000000000000004e-5 < z < 3.29999999999999988e-16

   1. Initial program 85.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 N/A

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

      15. /-lowering-/.f64 56.2

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

   5. Simplified 56.2%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 51.8

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

   8. Simplified 51.8%

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

Alternative 7: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.6e-70) t_1 (if (<= z 3.6e-128) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.6e-70) {
		tmp = t_1;
	} else if (z <= 3.6e-128) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-6.6d-70)) then
        tmp = t_1
    else if (z <= 3.6d-128) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.6e-70) {
		tmp = t_1;
	} else if (z <= 3.6e-128) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -6.6e-70:
		tmp = t_1
	elif z <= 3.6e-128:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.6e-70)
		tmp = t_1;
	elseif (z <= 3.6e-128)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -6.6e-70)
		tmp = t_1;
	elseif (z <= 3.6e-128)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e-70], t$95$1, If[LessEqual[z, 3.6e-128], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.60000000000000033e-70 or 3.60000000000000025e-128 < z

   1. Initial program 57.9%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 73.6

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

   5. Simplified 73.6%

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

   if -6.60000000000000033e-70 < z < 3.60000000000000025e-128

   1. Initial program 82.7%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 50.2%

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

Alternative 8: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+14}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.4e+57) t_1 (if (<= y 2.35e+14) (/ (- 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 <= -2.4e+57) {
		tmp = t_1;
	} else if (y <= 2.35e+14) {
		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 <= (-2.4d+57)) then
        tmp = t_1
    else if (y <= 2.35d+14) 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 <= -2.4e+57) {
		tmp = t_1;
	} else if (y <= 2.35e+14) {
		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 <= -2.4e+57:
		tmp = t_1
	elif y <= 2.35e+14:
		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 <= -2.4e+57)
		tmp = t_1;
	elseif (y <= 2.35e+14)
		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 <= -2.4e+57)
		tmp = t_1;
	elseif (y <= 2.35e+14)
		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, -2.4e+57], t$95$1, If[LessEqual[y, 2.35e+14], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000005e57 or 2.35e14 < y

   1. Initial program 51.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. /-lowering-/.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. --lowering--.f64 47.8

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

   5. Simplified 47.8%

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

   if -2.40000000000000005e57 < y < 2.35e14

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 56.1

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

   5. Simplified 56.1%

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

Alternative 9: 32.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 - z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (1.0 - 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 = x / (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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. /-lowering-/.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. --lowering--.f64 28.0

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

5. Simplified 28.0%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]
6. Add Preprocessing

Alternative 10: 24.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. associate--r+ N/A

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

   4. div-sub N/A

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

   5. --lowering--.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. associate-/l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. div-sub N/A

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

   11. sub-neg N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. +-lowering-+.f64 N/A

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

   15. /-lowering-/.f64 31.1

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

5. Simplified 31.1%

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

6. Taylor expanded in y around inf

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

8. Simplified 22.7%

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

Alternative 11: 24.2% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 66.4%

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

3. Taylor expanded in z around 0

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

5. Simplified 22.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 73.2% 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 2024204 
(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)))))