Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Percentage Accurate: 99.8% → 99.8%

Time: 16.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (fma (log y) x (+ (+ z t) (fma (+ b -0.5) (log c) a))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(log(y), x, ((z + t) + fma((b + -0.5), log(c), a))) + (y * i);
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(fma(log(y), x, Float64(Float64(z + t) + fma(Float64(b + -0.5), log(c), a))) + Float64(y * i))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[(z + t), $MachinePrecision] + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

   11. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

   12. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   13. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

   15. lower-fma.f64 99.5

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{\mathsf{fma}\left(b - 0.5, \log c, a\right)}\right) + y \cdot i \]

   16. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b - \frac{1}{2}}, \log c, a\right)\right) + y \cdot i \]

   17. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log c, a\right)\right) + y \cdot i \]

   18. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log c, a\right)\right) + y \cdot i \]

   19. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(b + \color{blue}{-0.5}, \log c, a\right)\right) + y \cdot i \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right)} + y \cdot i \]
5. Add Preprocessing

Alternative 2: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := y \cdot i + \left(\left(a + \left(t + \left(z + t\_1\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+29}:\\ \;\;\;\;a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + t\_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* y i) (+ (+ a (+ t (+ z t_1))) (* (log c) (- b 0.5))))))
   (if (<= t_2 (- INFINITY))
     (fma y i (* b (log c)))
     (if (<= t_2 1e+29)
       (+ a (fma (log c) (+ b -0.5) z))
       (if (<= t_2 2e+295) (+ a t_1) (+ a (* y i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = (y * i) + ((a + (t + (z + t_1))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(y, i, (b * log(c)));
	} else if (t_2 <= 1e+29) {
		tmp = a + fma(log(c), (b + -0.5), z);
	} else if (t_2 <= 2e+295) {
		tmp = a + t_1;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + t_1))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(y, i, Float64(b * log(c)));
	elseif (t_2 <= 1e+29)
		tmp = Float64(a + fma(log(c), Float64(b + -0.5), z));
	elseif (t_2 <= 2e+295)
		tmp = Float64(a + t_1);
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+29], N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+295], N[(a + t$95$1), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \log c} + y \cdot i \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]

      3. lower-log.f64 100.0

         \[\leadsto \color{blue}{\log c} \cdot b + y \cdot i \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\log c \cdot b + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + \log c \cdot b} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + \log c \cdot b \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \log c \cdot b\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, b \cdot \log c\right)} \]

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999914e28

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 80.8

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 70.3%

      \[\leadsto a + \mathsf{fma}\left(x, \color{blue}{\log y}, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto a + \left(z + \log c \cdot \color{blue}{\left(b - \frac{1}{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 49.2%

       \[\leadsto a + \mathsf{fma}\left(\log c, b + \color{blue}{-0.5}, z\right) \]

   if 9.99999999999999914e28 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2e295

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 83.7

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

      \[\leadsto a + x \cdot \color{blue}{\log y} \]

   if 2e295 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 97.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 91.4

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto a + i \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.6%

      \[\leadsto a + y \cdot \color{blue}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 10^{+29}:\\ \;\;\;\;a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := y \cdot i + \left(\left(a + \left(t + \left(z + t\_1\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + t\_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* y i) (+ (+ a (+ t (+ z t_1))) (* (log c) (- b 0.5))))))
   (if (<= t_2 -5e+67)
     (fma z (/ (* y i) z) z)
     (if (<= t_2 1e+29)
       (fma y i (* b (log c)))
       (if (<= t_2 2e+295) (+ a t_1) (+ a (* y i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = (y * i) + ((a + (t + (z + t_1))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_2 <= -5e+67) {
		tmp = fma(z, ((y * i) / z), z);
	} else if (t_2 <= 1e+29) {
		tmp = fma(y, i, (b * log(c)));
	} else if (t_2 <= 2e+295) {
		tmp = a + t_1;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + t_1))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_2 <= -5e+67)
		tmp = fma(z, Float64(Float64(y * i) / z), z);
	elseif (t_2 <= 1e+29)
		tmp = fma(y, i, Float64(b * log(c)));
	elseif (t_2 <= 2e+295)
		tmp = Float64(a + t_1);
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+67], N[(z * N[(N[(y * i), $MachinePrecision] / z), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$2, 1e+29], N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+295], N[(a + t$95$1), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.99999999999999976e67

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\right)}{z}, z\right)} \]

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 29.6%

      \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right) \]

   if -4.99999999999999976e67 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999914e28

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \log c} + y \cdot i \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]

      3. lower-log.f64 61.8

         \[\leadsto \color{blue}{\log c} \cdot b + y \cdot i \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\log c \cdot b} + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\log c \cdot b + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + \log c \cdot b} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + \log c \cdot b \]

      4. lower-fma.f64 61.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \log c \cdot b\right)} \]

   7. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, b \cdot \log c\right)} \]

   if 9.99999999999999914e28 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2e295

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 83.7

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

      \[\leadsto a + x \cdot \color{blue}{\log y} \]

   if 2e295 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 97.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 91.4

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto a + i \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.6%

      \[\leadsto a + y \cdot \color{blue}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := y \cdot i + \left(\left(a + \left(t + \left(z + t\_1\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + t\_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* y i) (+ (+ a (+ t (+ z t_1))) (* (log c) (- b 0.5))))))
   (if (<= t_2 1e+29)
     (fma z (/ (* y i) z) z)
     (if (<= t_2 2e+295) (+ a t_1) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = (y * i) + ((a + (t + (z + t_1))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_2 <= 1e+29) {
		tmp = fma(z, ((y * i) / z), z);
	} else if (t_2 <= 2e+295) {
		tmp = a + t_1;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + t_1))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_2 <= 1e+29)
		tmp = fma(z, Float64(Float64(y * i) / z), z);
	elseif (t_2 <= 2e+295)
		tmp = Float64(a + t_1);
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e+29], N[(z * N[(N[(y * i), $MachinePrecision] / z), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$2, 2e+295], N[(a + t$95$1), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999914e28

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\right)}{z}, z\right)} \]

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 28.2%

      \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right) \]

   if 9.99999999999999914e28 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2e295

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 83.7

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

      \[\leadsto a + x \cdot \color{blue}{\log y} \]

   if 2e295 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 97.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 91.4

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto a + i \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.6%

      \[\leadsto a + y \cdot \color{blue}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;a + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))))
   (if (<= t_1 (- INFINITY))
     (* y i)
     (if (<= t_1 100.0) (fma z (/ t z) z) (+ (+ t a) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 100.0) {
		tmp = fma(z, (t / z), z);
	} else {
		tmp = (t + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 100.0)
		tmp = fma(z, Float64(t / z), z);
	else
		tmp = Float64(Float64(t + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(z * N[(t / z), $MachinePrecision] + z), $MachinePrecision], N[(N[(t + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 87.9

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 100

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\right)}{z}, z\right)} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 28.6%

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

   if 100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 23.6

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{i \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      16. lower-+.f64 82.2

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

   8. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 52.8%

       \[\leadsto i \cdot y + \left(\color{blue}{t} + a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 100:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))))
   (if (<= t_1 (- INFINITY))
     (* y i)
     (if (<= t_1 200.0) (fma z (/ a z) z) (+ (+ t a) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 200.0) {
		tmp = fma(z, (a / z), z);
	} else {
		tmp = (t + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 200.0)
		tmp = fma(z, Float64(a / z), z);
	else
		tmp = Float64(Float64(t + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(z * N[(a / z), $MachinePrecision] + z), $MachinePrecision], N[(N[(t + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 87.9

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\right)}{z}, z\right)} \]

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 21.3%

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

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 23.9

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 23.9%

      \[\leadsto \color{blue}{i \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      16. lower-+.f64 81.9

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

   8. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.5%

       \[\leadsto i \cdot y + \left(\color{blue}{t} + a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 200:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -5e+29)
   (fma z (/ (* y i) z) z)
   (+ (+ t a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+29) {
		tmp = fma(z, ((y * i) / z), z);
	} else {
		tmp = (t + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -5e+29)
		tmp = fma(z, Float64(Float64(y * i) / z), z);
	else
		tmp = Float64(Float64(t + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+29], N[(z * N[(N[(y * i), $MachinePrecision] / z), $MachinePrecision] + z), $MachinePrecision], N[(N[(t + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000001e29

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\right)}{z}, z\right)} \]

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 28.7%

      \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right) \]

   if -5.0000000000000001e29 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 23.5

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 23.5%

      \[\leadsto \color{blue}{i \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      16. lower-+.f64 82.3

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

   8. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 52.5%

       \[\leadsto i \cdot y + \left(\color{blue}{t} + a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot i}{z}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + a\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ a (fma i y (fma (log c) (+ b -0.5) (fma x (log y) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a + fma(i, y, fma(log(c), (b + -0.5), fma(x, log(y), z)));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(a + fma(i, y, fma(log(c), Float64(b + -0.5), fma(x, log(y), z))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(a + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

   6. associate-+r+ N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

   9. lower-log.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

   10. sub-neg N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   13. +-commutative N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

   15. lower-log.f64 84.5

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

5. Applied rewrites 84.5%

   \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]
6. Add Preprocessing

Alternative 9: 94.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(-\left(-a\right)\right)\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (fma (log y) x (+ (+ z t) (- (- a)))))))
   (if (<= x -1.15e+86)
     t_1
     (if (<= x 1.8e+168)
       (+ (fma y i (fma (log c) (+ b -0.5) z)) (+ t a))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + fma(log(y), x, ((z + t) + -(-a)));
	double tmp;
	if (x <= -1.15e+86) {
		tmp = t_1;
	} else if (x <= 1.8e+168) {
		tmp = fma(y, i, fma(log(c), (b + -0.5), z)) + (t + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + fma(log(y), x, Float64(Float64(z + t) + Float64(-Float64(-a)))))
	tmp = 0.0
	if (x <= -1.15e+86)
		tmp = t_1;
	elseif (x <= 1.8e+168)
		tmp = Float64(fma(y, i, fma(log(c), Float64(b + -0.5), z)) + Float64(t + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(N[(z + t), $MachinePrecision] + (-(-a))), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+86], t$95$1, If[LessEqual[x, 1.8e+168], N[(N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14999999999999995e86 or 1.8e168 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

      12. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

      15. lower-fma.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{\mathsf{fma}\left(b - 0.5, \log c, a\right)}\right) + y \cdot i \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b - \frac{1}{2}}, \log c, a\right)\right) + y \cdot i \]

      17. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log c, a\right)\right) + y \cdot i \]

      18. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(\color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log c, a\right)\right) + y \cdot i \]

      19. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(b + \color{blue}{-0.5}, \log c, a\right)\right) + y \cdot i \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right)} + y \cdot i \]

   5. Taylor expanded in a around -inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{a} - 1\right)\right)}\right) + y \cdot i \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 91.9

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(-a \cdot \mathsf{fma}\left(\log c, -\frac{\color{blue}{b + -0.5}}{a}, -1\right)\right)\right) + y \cdot i \]

   7. Applied rewrites 91.9%

      \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \color{blue}{\left(-a \cdot \mathsf{fma}\left(\log c, -\frac{b + -0.5}{a}, -1\right)\right)}\right) + y \cdot i \]

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(\mathsf{neg}\left(-1 \cdot a\right)\right)\right) + y \cdot i \]
   9. Step-by-step derivation

   10. Applied rewrites 95.3%

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(-\left(-a\right)\right)\right) + y \cdot i \]

   if -1.14999999999999995e86 < x < 1.8e168

   1. Initial program 99.3%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 24.0

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 24.0%

      \[\leadsto \color{blue}{i \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      16. lower-+.f64 97.3

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

   8. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+86}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(-\left(-a\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(-\left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \log y \cdot x\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (* (log y) x))))
   (if (<= x -4.1e+254)
     t_1
     (if (<= x 5.2e+211)
       (+ (fma y i (fma (log c) (+ b -0.5) z)) (+ t a))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + (log(y) * x);
	double tmp;
	if (x <= -4.1e+254) {
		tmp = t_1;
	} else if (x <= 5.2e+211) {
		tmp = fma(y, i, fma(log(c), (b + -0.5), z)) + (t + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -4.1e+254)
		tmp = t_1;
	elseif (x <= 5.2e+211)
		tmp = Float64(fma(y, i, fma(log(c), Float64(b + -0.5), z)) + Float64(t + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+254], t$95$1, If[LessEqual[x, 5.2e+211], N[(N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999987e254 or 5.1999999999999997e211 < x

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. lower-log.f64 90.4

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -4.09999999999999987e254 < x < 5.1999999999999997e211

   1. Initial program 99.4%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 24.2

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 24.2%

      \[\leadsto \color{blue}{i \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

      16. lower-+.f64 92.7

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

   8. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \log y \cdot x\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (* (log y) x))))
   (if (<= x -4.1e+254)
     t_1
     (if (<= x 5.2e+211)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + (log(y) * x);
	double tmp;
	if (x <= -4.1e+254) {
		tmp = t_1;
	} else if (x <= 5.2e+211) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -4.1e+254)
		tmp = t_1;
	elseif (x <= 5.2e+211)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+254], t$95$1, If[LessEqual[x, 5.2e+211], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999987e254 or 5.1999999999999997e211 < x

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. lower-log.f64 90.4

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -4.09999999999999987e254 < x < 5.1999999999999997e211

   1. Initial program 99.4%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto a + \left(\color{blue}{\mathsf{fma}\left(i, y, z\right)} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + t\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, t\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, t\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, t\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, t\right)\right) \]

      12. lower-+.f64 92.7

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, t\right)\right) \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \log y \cdot x\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (* (log y) x))))
   (if (<= x -4.1e+254)
     t_1
     (if (<= x 5.2e+211) (+ a (fma i y (fma (log c) (+ b -0.5) z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + (log(y) * x);
	double tmp;
	if (x <= -4.1e+254) {
		tmp = t_1;
	} else if (x <= 5.2e+211) {
		tmp = a + fma(i, y, fma(log(c), (b + -0.5), z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -4.1e+254)
		tmp = t_1;
	elseif (x <= 5.2e+211)
		tmp = Float64(a + fma(i, y, fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+254], t$95$1, If[LessEqual[x, 5.2e+211], N[(a + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999987e254 or 5.1999999999999997e211 < x

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. lower-log.f64 90.4

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -4.09999999999999987e254 < x < 5.1999999999999997e211

   1. Initial program 99.4%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 82.8

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto a + \mathsf{fma}\left(i, y, z + \log c \cdot \left(b - \frac{1}{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+254}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.0% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \left(t + a\right) + y \cdot i \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(t + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 22.3

      \[\leadsto \color{blue}{i \cdot y} \]

5. Applied rewrites 22.3%

   \[\leadsto \color{blue}{i \cdot y} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + \left(t + a\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + \left(t + a\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + \left(t + a\right) \]

   12. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + \left(t + a\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + \left(t + a\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

   15. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z\right)\right) + \left(t + a\right) \]

   16. lower-+.f64 81.6

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \color{blue}{\left(t + a\right)} \]

8. Applied rewrites 81.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + \left(t + a\right)} \]

9. Taylor expanded in y around inf

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

11. Applied rewrites 52.1%

    \[\leadsto i \cdot y + \left(\color{blue}{t} + a\right) \]

12. Final simplification 52.1%

    \[\leadsto \left(t + a\right) + y \cdot i \]
13. Add Preprocessing

Alternative 14: 38.4% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ a + y \cdot i \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

   6. associate-+r+ N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

   9. lower-log.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

   10. sub-neg N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   13. +-commutative N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

   15. lower-log.f64 84.5

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

5. Applied rewrites 84.5%

   \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

6. Taylor expanded in i around inf

   \[\leadsto a + i \cdot \color{blue}{y} \]
7. Step-by-step derivation

8. Applied rewrites 38.1%

   \[\leadsto a + y \cdot \color{blue}{i} \]
9. Add Preprocessing

Alternative 15: 24.4% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ y \cdot i \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 22.3

      \[\leadsto \color{blue}{i \cdot y} \]

5. Applied rewrites 22.3%

   \[\leadsto \color{blue}{i \cdot y} \]

6. Final simplification 22.3%

   \[\leadsto y \cdot i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))