Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.6% → 94.4%

Time: 15.1s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 94.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-270}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-301}:\\ \;\;\;\;t\_1 - \frac{\mathsf{fma}\left(-x, \frac{y}{b - y}, \frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(b - y\right)}\right)}{z}\\ \mathbf{elif}\;t\_3 \leq 10^{+285}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma z (- b y) y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma z (/ (- t a) t_2) (* x (/ y t_2)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-270)
       t_3
       (if (<= t_3 5e-301)
         (-
          t_1
          (/
           (fma (- x) (/ y (- b y)) (/ (* y (- t a)) (* (- b y) (- b y))))
           z))
         (if (<= t_3 1e+285) t_3 (if (<= t_3 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma(z, (b - y), y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-270) {
		tmp = t_3;
	} else if (t_3 <= 5e-301) {
		tmp = t_1 - (fma(-x, (y / (b - y)), ((y * (t - a)) / ((b - y) * (b - y)))) / z);
	} else if (t_3 <= 1e+285) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(z, Float64(b - y), y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-270)
		tmp = t_3;
	elseif (t_3 <= 5e-301)
		tmp = Float64(t_1 - Float64(fma(Float64(-x), Float64(y / Float64(b - y)), Float64(Float64(y * Float64(t - a)) / Float64(Float64(b - y) * Float64(b - y)))) / z));
	elseif (t_3 <= 1e+285)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-270], t$95$3, If[LessEqual[t$95$3, 5e-301], N[(t$95$1 - N[(N[((-x) * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+285], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

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 or 9.9999999999999998e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Simplified 99.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2.0000000000000001e-270 or 5.00000000000000013e-301 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999998e284

   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

   if -2.0000000000000001e-270 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.00000000000000013e-301

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 44.8

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

   5. Simplified 44.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 44.8

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

   7. Applied egg-rr 44.8%

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

   8. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

   10. Simplified 90.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 87.5

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

   5. Simplified 87.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-270}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-x, \frac{y}{b - y}, \frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(b - y\right)}\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{t\_1}, x \cdot \frac{y}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+285}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (/ (- t a) t_1) (* x (/ y t_1)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 1e+285) t_2 (if (<= t_2 INFINITY) t_3 (/ (- t a) (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, ((t - a) / t_1), (x * (y / t_1)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e+285) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(Float64(t - a) / t_1), Float64(x * Float64(y / t_1)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e+285)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 1e+285], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 9.9999999999999998e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Simplified 99.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999998e284

   1. Initial program 92.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 87.5

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

   5. Simplified 87.5%

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

Alternative 3: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma z (/ (- t a) (fma z (- b y) y)) x)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+285) t_1 (if (<= t_1 INFINITY) t_2 (/ (- t a) (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma(z, ((t - a) / fma(z, (b - y), y)), x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+285) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+285)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+285], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 9.9999999999999998e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 86.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999998e284

   1. Initial program 92.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 87.5

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

   5. Simplified 87.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{t\_1}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -a, x \cdot y\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -9.2e+24)
     t_2
     (if (<= z -8.2e-38)
       (/ (fma z t (* x y)) t_1)
       (if (<= z -6.5e-96)
         (/ (fma x y (* z (- a))) t_1)
         (if (<= z 3.9e-142)
           (fma z (/ (- t a) y) x)
           (if (<= z 8.8e+27) (/ (fma z (- a) (* x y)) t_1) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -9.2e+24) {
		tmp = t_2;
	} else if (z <= -8.2e-38) {
		tmp = fma(z, t, (x * y)) / t_1;
	} else if (z <= -6.5e-96) {
		tmp = fma(x, y, (z * -a)) / t_1;
	} else if (z <= 3.9e-142) {
		tmp = fma(z, ((t - a) / y), x);
	} else if (z <= 8.8e+27) {
		tmp = fma(z, -a, (x * y)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9.2e+24)
		tmp = t_2;
	elseif (z <= -8.2e-38)
		tmp = Float64(fma(z, t, Float64(x * y)) / t_1);
	elseif (z <= -6.5e-96)
		tmp = Float64(fma(x, y, Float64(z * Float64(-a))) / t_1);
	elseif (z <= 3.9e-142)
		tmp = fma(z, Float64(Float64(t - a) / y), x);
	elseif (z <= 8.8e+27)
		tmp = Float64(fma(z, Float64(-a), Float64(x * y)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+24], t$95$2, If[LessEqual[z, -8.2e-38], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, -6.5e-96], N[(N[(x * y + N[(z * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 3.9e-142], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.8e+27], N[(N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -9.1999999999999996e24 or 8.7999999999999995e27 < 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 84.2

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

   5. Simplified 84.2%

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

   if -9.1999999999999996e24 < z < -8.1999999999999996e-38

   1. Initial program 92.8%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 86.0

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

   5. Simplified 86.0%

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

   if -8.1999999999999996e-38 < z < -6.50000000000000001e-96

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 91.0

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

   7. Simplified 91.0%

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

   if -6.50000000000000001e-96 < z < 3.9000000000000003e-142

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 94.2

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

   5. Simplified 94.2%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 90.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 78.7

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

   11. Simplified 78.7%

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

   if 3.9000000000000003e-142 < z < 8.7999999999999995e27

   1. Initial program 84.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 63.9

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

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, -a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{t\_1}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (fma x y (* z (- a))) t_1)))
   (if (<= z -9.2e+24)
     t_2
     (if (<= z -8.2e-38)
       (/ (fma z t (* x y)) t_1)
       (if (<= z -6.5e-96)
         t_3
         (if (<= z 3.9e-142)
           (fma z (/ (- t a) y) x)
           (if (<= z 8.8e+27) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double t_3 = fma(x, y, (z * -a)) / t_1;
	double tmp;
	if (z <= -9.2e+24) {
		tmp = t_2;
	} else if (z <= -8.2e-38) {
		tmp = fma(z, t, (x * y)) / t_1;
	} else if (z <= -6.5e-96) {
		tmp = t_3;
	} else if (z <= 3.9e-142) {
		tmp = fma(z, ((t - a) / y), x);
	} else if (z <= 8.8e+27) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(fma(x, y, Float64(z * Float64(-a))) / t_1)
	tmp = 0.0
	if (z <= -9.2e+24)
		tmp = t_2;
	elseif (z <= -8.2e-38)
		tmp = Float64(fma(z, t, Float64(x * y)) / t_1);
	elseif (z <= -6.5e-96)
		tmp = t_3;
	elseif (z <= 3.9e-142)
		tmp = fma(z, Float64(Float64(t - a) / y), x);
	elseif (z <= 8.8e+27)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * y + N[(z * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[z, -9.2e+24], t$95$2, If[LessEqual[z, -8.2e-38], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, -6.5e-96], t$95$3, If[LessEqual[z, 3.9e-142], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.8e+27], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.1999999999999996e24 or 8.7999999999999995e27 < 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 84.2

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

   5. Simplified 84.2%

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

   if -9.1999999999999996e24 < z < -8.1999999999999996e-38

   1. Initial program 92.8%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 86.0

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

   5. Simplified 86.0%

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

   if -8.1999999999999996e-38 < z < -6.50000000000000001e-96 or 3.9000000000000003e-142 < z < 8.7999999999999995e27

   1. Initial program 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 69.4

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

   7. Simplified 69.4%

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

   if -6.50000000000000001e-96 < z < 3.9000000000000003e-142

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 94.2

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

   5. Simplified 94.2%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 90.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 78.7

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

   11. Simplified 78.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{t\_1}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_1}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -9.2e+24)
     t_2
     (if (<= z -8.2e-38)
       (/ (fma z t (* x y)) t_1)
       (if (<= z -2.2e-94)
         (/ (fma x y (* z (- a))) t_1)
         (if (<= z 7e-14)
           (fma z (/ (- t a) t_1) x)
           (if (<= z 8.8e+27) (* x (/ y t_1)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -9.2e+24) {
		tmp = t_2;
	} else if (z <= -8.2e-38) {
		tmp = fma(z, t, (x * y)) / t_1;
	} else if (z <= -2.2e-94) {
		tmp = fma(x, y, (z * -a)) / t_1;
	} else if (z <= 7e-14) {
		tmp = fma(z, ((t - a) / t_1), x);
	} else if (z <= 8.8e+27) {
		tmp = x * (y / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9.2e+24)
		tmp = t_2;
	elseif (z <= -8.2e-38)
		tmp = Float64(fma(z, t, Float64(x * y)) / t_1);
	elseif (z <= -2.2e-94)
		tmp = Float64(fma(x, y, Float64(z * Float64(-a))) / t_1);
	elseif (z <= 7e-14)
		tmp = fma(z, Float64(Float64(t - a) / t_1), x);
	elseif (z <= 8.8e+27)
		tmp = Float64(x * Float64(y / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+24], t$95$2, If[LessEqual[z, -8.2e-38], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, -2.2e-94], N[(N[(x * y + N[(z * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 7e-14], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.8e+27], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -9.1999999999999996e24 or 8.7999999999999995e27 < 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 84.2

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

   5. Simplified 84.2%

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

   if -9.1999999999999996e24 < z < -8.1999999999999996e-38

   1. Initial program 92.8%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 86.0

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

   5. Simplified 86.0%

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

   if -8.1999999999999996e-38 < z < -2.20000000000000001e-94

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 91.0

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

   7. Simplified 91.0%

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

   if -2.20000000000000001e-94 < z < 7.0000000000000005e-14

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 86.9%

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

   if 7.0000000000000005e-14 < z < 8.7999999999999995e27

   1. Initial program 59.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 65.7

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

   5. Simplified 65.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_2}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))) (t_2 (fma z (- b y) y)))
   (if (<= z -8.6e-44)
     t_1
     (if (<= z 5e-131)
       (fma z (/ (- t a) y) x)
       (if (<= z 1.1e-38)
         (/ (* z (- t a)) t_2)
         (if (<= z 8.8e+27) (* x (/ y t_2)) 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 t_2 = fma(z, (b - y), y);
	double tmp;
	if (z <= -8.6e-44) {
		tmp = t_1;
	} else if (z <= 5e-131) {
		tmp = fma(z, ((t - a) / y), x);
	} else if (z <= 1.1e-38) {
		tmp = (z * (t - a)) / t_2;
	} else if (z <= 8.8e+27) {
		tmp = x * (y / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (z <= -8.6e-44)
		tmp = t_1;
	elseif (z <= 5e-131)
		tmp = fma(z, Float64(Float64(t - a) / y), x);
	elseif (z <= 1.1e-38)
		tmp = Float64(Float64(z * Float64(t - a)) / t_2);
	elseif (z <= 8.8e+27)
		tmp = Float64(x * Float64(y / t_2));
	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]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -8.6e-44], t$95$1, If[LessEqual[z, 5e-131], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.1e-38], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[z, 8.8e+27], N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.60000000000000027e-44 or 8.7999999999999995e27 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in z around inf

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

      1. 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. Simplified 80.4%

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

   if -8.60000000000000027e-44 < z < 5.0000000000000004e-131

   1. Initial program 85.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 87.2%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 75.2

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

   11. Simplified 75.2%

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

   if 5.0000000000000004e-131 < z < 1.10000000000000004e-38

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 65.0

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

   5. Simplified 65.0%

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

   if 1.10000000000000004e-38 < z < 8.7999999999999995e27

   1. Initial program 63.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 68.1

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

   5. Simplified 68.1%

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

Alternative 8: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8.6e-44)
     t_1
     (if (<= z 2.5e-40)
       (fma z (/ (- t a) y) x)
       (if (<= z 8.8e+27) (* x (/ y (fma z (- b y) 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 <= -8.6e-44) {
		tmp = t_1;
	} else if (z <= 2.5e-40) {
		tmp = fma(z, ((t - a) / y), x);
	} else if (z <= 8.8e+27) {
		tmp = x * (y / fma(z, (b - y), 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 <= -8.6e-44)
		tmp = t_1;
	elseif (z <= 2.5e-40)
		tmp = fma(z, Float64(Float64(t - a) / y), x);
	elseif (z <= 8.8e+27)
		tmp = Float64(x * Float64(y / fma(z, Float64(b - y), 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, -8.6e-44], t$95$1, If[LessEqual[z, 2.5e-40], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.8e+27], N[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.60000000000000027e-44 or 8.7999999999999995e27 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in z around inf

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

      1. 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. Simplified 80.4%

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

   if -8.60000000000000027e-44 < z < 2.49999999999999982e-40

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 84.7%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 68.9

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

   11. Simplified 68.9%

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

   if 2.49999999999999982e-40 < z < 8.7999999999999995e27

   1. Initial program 65.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 64.2

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

   5. Simplified 64.2%

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

Alternative 9: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.2e+24)
     t_1
     (if (<= z 8.8e+27) (/ (fma z t (* x y)) (fma z (- b y) 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 <= -9.2e+24) {
		tmp = t_1;
	} else if (z <= 8.8e+27) {
		tmp = fma(z, t, (x * y)) / fma(z, (b - y), 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 <= -9.2e+24)
		tmp = t_1;
	elseif (z <= 8.8e+27)
		tmp = Float64(fma(z, t, Float64(x * y)) / fma(z, Float64(b - y), 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, -9.2e+24], t$95$1, If[LessEqual[z, 8.8e+27], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999996e24 or 8.7999999999999995e27 < 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 84.2

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

   5. Simplified 84.2%

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

   if -9.1999999999999996e24 < z < 8.7999999999999995e27

   1. Initial program 85.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 66.8

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

   5. Simplified 66.8%

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

Alternative 10: 37.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(x, z, x\right), x\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+212}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+288}:\\ \;\;\;\;-\frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.3e-35)
   (/ t b)
   (if (<= z 3e-27)
     (fma z (fma x z x) x)
     (if (<= z 5.1e+212)
       (- (/ a b))
       (if (<= z 1.05e+288) (- (/ t y)) (/ t b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.3e-35) {
		tmp = t / b;
	} else if (z <= 3e-27) {
		tmp = fma(z, fma(x, z, x), x);
	} else if (z <= 5.1e+212) {
		tmp = -(a / b);
	} else if (z <= 1.05e+288) {
		tmp = -(t / y);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.3e-35)
		tmp = Float64(t / b);
	elseif (z <= 3e-27)
		tmp = fma(z, fma(x, z, x), x);
	elseif (z <= 5.1e+212)
		tmp = Float64(-Float64(a / b));
	elseif (z <= 1.05e+288)
		tmp = Float64(-Float64(t / y));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.3e-35], N[(t / b), $MachinePrecision], If[LessEqual[z, 3e-27], N[(z * N[(x * z + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.1e+212], (-N[(a / b), $MachinePrecision]), If[LessEqual[z, 1.05e+288], (-N[(t / y), $MachinePrecision]), N[(t / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.3000000000000002e-35 or 1.04999999999999999e288 < z

   1. Initial program 48.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 26.7

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

   5. Simplified 26.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 37.5

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

   8. Simplified 37.5%

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

   if -4.3000000000000002e-35 < z < 3.0000000000000001e-27

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 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 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 47.3

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

   8. Simplified 47.3%

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

   if 3.0000000000000001e-27 < z < 5.1000000000000002e212

   1. Initial program 64.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.0

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

   5. Simplified 34.0%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{b}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{b}\right)} \]

      3. lower-/.f64 34.4

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

   8. Simplified 34.4%

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

   if 5.1000000000000002e212 < z < 1.04999999999999999e288

   1. Initial program 30.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 20.7

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

   5. Simplified 20.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 20.7

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

   8. Simplified 20.7%

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

   9. Taylor expanded in b around 0

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

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{t}{y}\right)} \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\frac{t}{\mathsf{neg}\left(y\right)}} \]

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

          \[\leadsto \frac{t}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

       6. lower-neg.f64 40.2

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

   11. Simplified 40.2%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(x, z, x\right), x\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+212}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+288}:\\ \;\;\;\;-\frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8.6e-44) t_1 (if (<= z 7e-27) (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 <= -8.6e-44) {
		tmp = t_1;
	} else if (z <= 7e-27) {
		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 <= -8.6e-44)
		tmp = t_1;
	elseif (z <= 7e-27)
		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, -8.6e-44], t$95$1, If[LessEqual[z, 7e-27], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.60000000000000027e-44 or 7.0000000000000003e-27 < z

   1. Initial program 50.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 74.9

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

   5. Simplified 74.9%

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

   if -8.60000000000000027e-44 < z < 7.0000000000000003e-27

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 84.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 68.3

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

   11. Simplified 68.3%

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

Alternative 12: 34.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.95 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-58}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.95e-26)
   x
   (if (<= y 1.75e-58) (- (/ a b)) (if (<= y 1.3e+64) (/ t b) (fma x z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.95e-26) {
		tmp = x;
	} else if (y <= 1.75e-58) {
		tmp = -(a / b);
	} else if (y <= 1.3e+64) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.95e-26)
		tmp = x;
	elseif (y <= 1.75e-58)
		tmp = Float64(-Float64(a / b));
	elseif (y <= 1.3e+64)
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.95e-26], x, If[LessEqual[y, 1.75e-58], (-N[(a / b), $MachinePrecision]), If[LessEqual[y, 1.3e+64], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.94999999999999972e-26

   1. Initial program 56.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 52.0

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

   5. Simplified 52.0%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 34.3%

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

   if -3.94999999999999972e-26 < y < 1.75e-58

   1. Initial program 83.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{b}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{b}\right)} \]

      3. lower-/.f64 40.6

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

   8. Simplified 40.6%

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

   if 1.75e-58 < y < 1.29999999999999998e64

   1. Initial program 68.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 29.4

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

   5. Simplified 29.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 41.6

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

   8. Simplified 41.6%

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

   if 1.29999999999999998e64 < y

   1. Initial program 54.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.6

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

   8. Simplified 46.6%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.95 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-58}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e-87) t_1 (if (<= z 4e-142) (fma x z 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 <= -3.3e-87) {
		tmp = t_1;
	} else if (z <= 4e-142) {
		tmp = fma(x, z, 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 <= -3.3e-87)
		tmp = t_1;
	elseif (z <= 4e-142)
		tmp = fma(x, z, 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, -3.3e-87], t$95$1, If[LessEqual[z, 4e-142], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e-87 or 4.0000000000000002e-142 < z

   1. Initial program 60.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 67.6

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

   5. Simplified 67.6%

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

   if -3.3e-87 < z < 4.0000000000000002e-142

   1. Initial program 84.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 55.9

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

   5. Simplified 55.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 55.9

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

   8. Simplified 55.9%

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

Alternative 14: 53.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -0.88) t_1 (if (<= y 1.7e+64) (/ (- 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 <= -0.88) {
		tmp = t_1;
	} else if (y <= 1.7e+64) {
		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 <= (-0.88d0)) then
        tmp = t_1
    else if (y <= 1.7d+64) 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 <= -0.88) {
		tmp = t_1;
	} else if (y <= 1.7e+64) {
		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 <= -0.88:
		tmp = t_1
	elif y <= 1.7e+64:
		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 <= -0.88)
		tmp = t_1;
	elseif (y <= 1.7e+64)
		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 <= -0.88)
		tmp = t_1;
	elseif (y <= 1.7e+64)
		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, -0.88], t$95$1, If[LessEqual[y, 1.7e+64], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.880000000000000004 or 1.7000000000000001e64 < y

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 57.4

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

   5. Simplified 57.4%

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

   if -0.880000000000000004 < y < 1.7000000000000001e64

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 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. Simplified 59.2%

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

Alternative 15: 43.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.85e+64) t_1 (if (<= y 2.1e+64) (/ t (- b y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.85e+64) {
		tmp = t_1;
	} else if (y <= 2.1e+64) {
		tmp = t / (b - 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 = x / (1.0d0 - z)
    if (y <= (-1.85d+64)) then
        tmp = t_1
    else if (y <= 2.1d+64) then
        tmp = t / (b - 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 = x / (1.0 - z);
	double tmp;
	if (y <= -1.85e+64) {
		tmp = t_1;
	} else if (y <= 2.1e+64) {
		tmp = t / (b - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.85e+64:
		tmp = t_1
	elif y <= 2.1e+64:
		tmp = t / (b - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.85e+64)
		tmp = t_1;
	elseif (y <= 2.1e+64)
		tmp = Float64(t / Float64(b - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.85e+64)
		tmp = t_1;
	elseif (y <= 2.1e+64)
		tmp = t / (b - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e+64], t$95$1, If[LessEqual[y, 2.1e+64], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999992e64 or 2.1e64 < 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. 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 61.0

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

   5. Simplified 61.0%

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

   if -1.84999999999999992e64 < y < 2.1e64

   1. Initial program 80.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 34.4

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

   5. Simplified 34.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 37.3

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

   8. Simplified 37.3%

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

Alternative 16: 45.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.155:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(x, z, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.05e-78) t_1 (if (<= z 0.155) (fma z (fma x z x) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.05e-78) {
		tmp = t_1;
	} else if (z <= 0.155) {
		tmp = fma(z, fma(x, z, x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.05e-78)
		tmp = t_1;
	elseif (z <= 0.155)
		tmp = fma(z, fma(x, z, x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.05e-78], t$95$1, If[LessEqual[z, 0.155], N[(z * N[(x * z + x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0499999999999999e-78 or 0.154999999999999999 < z

   1. Initial program 51.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 23.6

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

   5. Simplified 23.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 41.6

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

   8. Simplified 41.6%

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

   if -2.0499999999999999e-78 < z < 0.154999999999999999

   1. Initial program 86.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 47.4

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

   5. Simplified 47.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 47.4

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

   8. Simplified 47.4%

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

Alternative 17: 38.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 0.155:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(x, z, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.3e-35) (/ t b) (if (<= z 0.155) (fma z (fma x z x) x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.3e-35) {
		tmp = t / b;
	} else if (z <= 0.155) {
		tmp = fma(z, fma(x, z, x), x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.3e-35)
		tmp = Float64(t / b);
	elseif (z <= 0.155)
		tmp = fma(z, fma(x, z, x), x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.3e-35], N[(t / b), $MachinePrecision], If[LessEqual[z, 0.155], N[(z * N[(x * z + x), $MachinePrecision] + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3000000000000002e-35 or 0.154999999999999999 < z

   1. Initial program 48.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 24.2

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

   5. Simplified 24.2%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 28.6

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

   8. Simplified 28.6%

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

   if -4.3000000000000002e-35 < z < 0.154999999999999999

   1. Initial program 87.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 45.8

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

   8. Simplified 45.8%

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

Alternative 18: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 0.155:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.3e-35) (/ t b) (if (<= z 0.155) (* x (+ z 1.0)) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.3e-35) {
		tmp = t / b;
	} else if (z <= 0.155) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t / 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 <= (-4.3d-35)) then
        tmp = t / b
    else if (z <= 0.155d0) then
        tmp = x * (z + 1.0d0)
    else
        tmp = t / 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 <= -4.3e-35) {
		tmp = t / b;
	} else if (z <= 0.155) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -4.3e-35:
		tmp = t / b
	elif z <= 0.155:
		tmp = x * (z + 1.0)
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.3e-35)
		tmp = Float64(t / b);
	elseif (z <= 0.155)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -4.3e-35)
		tmp = t / b;
	elseif (z <= 0.155)
		tmp = x * (z + 1.0);
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.3e-35], N[(t / b), $MachinePrecision], If[LessEqual[z, 0.155], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3000000000000002e-35 or 0.154999999999999999 < z

   1. Initial program 48.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 24.2

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

   5. Simplified 24.2%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 28.6

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

   8. Simplified 28.6%

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

   if -4.3000000000000002e-35 < z < 0.154999999999999999

   1. Initial program 87.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 45.8

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

   8. Simplified 45.8%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.8

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

   10. Applied egg-rr 45.8%

       \[\leadsto \color{blue}{\left(z + 1\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 0.155:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.8% 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 68.2%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 34.1

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

5. Simplified 34.1%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 25.8

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

8. Simplified 25.8%

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

Alternative 20: 3.8% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 34.1

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

5. Simplified 34.1%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 25.8

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

8. Simplified 25.8%

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

9. Taylor expanded in z around inf

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

    1. lower-*.f64 4.4

       \[\leadsto \color{blue}{x \cdot z} \]

11. Simplified 4.4%

    \[\leadsto \color{blue}{x \cdot z} \]
12. Add Preprocessing

Developer Target 1: 73.6% 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 2024208 
(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)))))