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

Percentage Accurate: 99.8% → 99.8%

Time: 14.1s

Alternatives: 18

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 45.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + i \cdot y\\ \mathbf{if}\;t\_1 \leq -1.9 \cdot 10^{+307}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* i y))))
   (if (<= t_1 -1.9e+307)
     (* i y)
     (if (<= t_1 -4e+27) (fma (/ z t) t t) (+ (* i y) (+ a t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (i * y);
	double tmp;
	if (t_1 <= -1.9e+307) {
		tmp = i * y;
	} else if (t_1 <= -4e+27) {
		tmp = fma((z / t), t, t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(i * y))
	tmp = 0.0
	if (t_1 <= -1.9e+307)
		tmp = Float64(i * y);
	elseif (t_1 <= -4e+27)
		tmp = fma(Float64(z / t), t, t);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.9e+307], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -4e+27], N[(N[(z / t), $MachinePrecision] * t + t), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $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)) < -1.90000000000000011e307

   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. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.90000000000000011e307 < (+.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)) < -4.0000000000000001e27

   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 -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 25.7%

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

   if -4.0000000000000001e27 < (+.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 99.8%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 50.8%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + i \cdot y \leq -1.9 \cdot 10^{+307}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + i \cdot y \leq -4 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;t\_1 \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= t_1 -4e+101)
     (+ (fma (- b 0.5) (log c) z) (+ a t))
     (if (<= t_1 -275.0)
       (+ (* x (log y)) (+ a t))
       (if (<= t_1 2e+166)
         (+ (fma -0.5 (log c) (fma i y z)) (+ a t))
         (fma y i (* b (log c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (t_1 <= -4e+101) {
		tmp = fma((b - 0.5), log(c), z) + (a + t);
	} else if (t_1 <= -275.0) {
		tmp = (x * log(y)) + (a + t);
	} else if (t_1 <= 2e+166) {
		tmp = fma(-0.5, log(c), fma(i, y, z)) + (a + t);
	} else {
		tmp = fma(y, i, (b * log(c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_1 <= -4e+101)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + Float64(a + t));
	elseif (t_1 <= -275.0)
		tmp = Float64(Float64(x * log(y)) + Float64(a + t));
	elseif (t_1 <= 2e+166)
		tmp = Float64(fma(-0.5, log(c), fma(i, y, z)) + Float64(a + t));
	else
		tmp = fma(y, i, Float64(b * log(c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+101], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -275.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+166], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -3.9999999999999999e101

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.7%

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

   if -3.9999999999999999e101 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -275

   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 y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.5%

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

   if -275 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 1.99999999999999988e166

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 83.1%

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

   if 1.99999999999999988e166 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 30.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 30.1

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

   10. Applied rewrites 30.1%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{b \cdot \log c}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-log.f64 88.1

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

   13. Applied rewrites 88.1%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{b \cdot \log c}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, i \cdot y\right)\right) + z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= t_1 -2e+120)
     (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ a t))
     (if (<= t_1 5e+85)
       (+ (+ (+ (fma -0.5 (log c) (fma (log y) x (* i y))) z) t) a)
       (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (t_1 <= -2e+120) {
		tmp = fma((b - 0.5), log(c), fma(log(y), x, z)) + (a + t);
	} else if (t_1 <= 5e+85) {
		tmp = ((fma(-0.5, log(c), fma(log(y), x, (i * y))) + z) + t) + a;
	} else {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_1 <= -2e+120)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, z)) + Float64(a + t));
	elseif (t_1 <= 5e+85)
		tmp = Float64(Float64(Float64(fma(-0.5, log(c), fma(log(y), x, Float64(i * y))) + z) + t) + a);
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+120], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+85], N[(N[(N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2e120

   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 y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -2e120 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 5.0000000000000001e85

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.0000000000000001e85 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, i \cdot y\right)\right) + z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, b \cdot \log c\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* b (log c)))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -4e+101)
     t_1
     (if (<= t_2 -275.0)
       (+ (* x (log y)) (+ a t))
       (if (<= t_2 2e+166) (+ (fma (/ z a) a a) (* i y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (b * log(c)));
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -4e+101) {
		tmp = t_1;
	} else if (t_2 <= -275.0) {
		tmp = (x * log(y)) + (a + t);
	} else if (t_2 <= 2e+166) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(b * log(c)))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -4e+101)
		tmp = t_1;
	elseif (t_2 <= -275.0)
		tmp = Float64(Float64(x * log(y)) + Float64(a + t));
	elseif (t_2 <= 2e+166)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+101], t$95$1, If[LessEqual[t$95$2, -275.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+166], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -3.9999999999999999e101 or 1.99999999999999988e166 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 31.3

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

   10. Applied rewrites 31.4%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{b \cdot \log c}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-log.f64 70.7

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

   13. Applied rewrites 70.7%

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

   if -3.9999999999999999e101 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -275

   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 y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.5%

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

   if -275 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 1.99999999999999988e166

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 56.7%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, a, a\right) + y \cdot i \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+101}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \mathbf{elif}\;t\_2 \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -4e+101)
     (+ t_1 (+ a t))
     (if (<= t_2 -275.0)
       (+ (* x (log y)) (+ a t))
       (if (<= t_2 1e+250) (+ (fma (/ z a) a a) (* i y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -4e+101) {
		tmp = t_1 + (a + t);
	} else if (t_2 <= -275.0) {
		tmp = (x * log(y)) + (a + t);
	} else if (t_2 <= 1e+250) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -4e+101)
		tmp = Float64(t_1 + Float64(a + t));
	elseif (t_2 <= -275.0)
		tmp = Float64(Float64(x * log(y)) + Float64(a + t));
	elseif (t_2 <= 1e+250)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+101], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -275.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+250], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -3.9999999999999999e101

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 72.2%

      \[\leadsto \left(t + a\right) + b \cdot \color{blue}{\log c} \]

   if -3.9999999999999999e101 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -275

   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 y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.5%

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

   if -275 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999992e249

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 57.1%

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

   if 9.9999999999999992e249 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 88.6

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

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{b \cdot \log c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -4 \cdot 10^{+101}:\\ \;\;\;\;b \cdot \log c + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq -275:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+120)
     (+ t_1 (+ a t))
     (if (<= t_2 1e+250) (+ (fma (/ z a) a a) (* i y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+120) {
		tmp = t_1 + (a + t);
	} else if (t_2 <= 1e+250) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+120)
		tmp = Float64(t_1 + Float64(a + t));
	elseif (t_2 <= 1e+250)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+120], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+250], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2e120

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 77.1%

      \[\leadsto \left(t + a\right) + b \cdot \color{blue}{\log c} \]

   if -2e120 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999992e249

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.1%

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

   if 9.9999999999999992e249 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 88.6

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

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{b \cdot \log c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+120}:\\ \;\;\;\;b \cdot \log c + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+120)
     t_1
     (if (<= t_2 1e+250) (+ (fma (/ z a) a a) (* i y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+120) {
		tmp = t_1;
	} else if (t_2 <= 1e+250) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+120)
		tmp = t_1;
	elseif (t_2 <= 1e+250)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+120], t$95$1, If[LessEqual[t$95$2, 1e+250], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2e120 or 9.9999999999999992e249 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2e120 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999992e249

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+120}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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[(i * y), $MachinePrecision]), $MachinePrecision], -100.0], N[(y * i + N[(N[(z / t), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $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)) < -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 t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 55.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 55.0

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

   10. Applied rewrites 55.1%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 40.0%

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

   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 99.8%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.2%

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

4. Final simplification 45.8%

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

Alternative 10: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ a t))))
   (if (<= x -2.1e+107)
     t_1
     (if (<= x 4e+88) (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a 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 = fma((b - 0.5), log(c), fma(log(y), x, z)) + (a + t);
	double tmp;
	if (x <= -2.1e+107) {
		tmp = t_1;
	} else if (x <= 4e+88) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, z)) + Float64(a + t))
	tmp = 0.0
	if (x <= -2.1e+107)
		tmp = t_1;
	elseif (x <= 4e+88)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + 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[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+107], t$95$1, If[LessEqual[x, 4e+88], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e107 or 3.99999999999999984e88 < 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. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if -2.1e107 < x < 3.99999999999999984e88

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, b \cdot \log c\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -2.4e+218)
   (fma y i (* x (log y)))
   (if (<= x 1.3e+155)
     (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
     (+ (fma (log y) x (* b (log c))) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -2.4e+218) {
		tmp = fma(y, i, (x * log(y)));
	} else if (x <= 1.3e+155) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = fma(log(y), x, (b * log(c))) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -2.4e+218)
		tmp = fma(y, i, Float64(x * log(y)));
	elseif (x <= 1.3e+155)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = Float64(fma(log(y), x, Float64(b * log(c))) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -2.4e+218], N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+155], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999981e218

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 62.2

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

   10. Applied rewrites 62.2%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 83.2

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

   13. Applied rewrites 83.2%

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

   if -2.39999999999999981e218 < x < 1.3000000000000001e155

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 1.3000000000000001e155 < 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. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 82.4

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

   7. Applied rewrites 82.4%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{b \cdot \log c}\right) + y \cdot i \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, b \cdot \log c\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, t\right) + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -2.4e+218)
   (fma y i (* x (log y)))
   (if (<= x 1.1e+156)
     (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
     (+ (fma (- b 0.5) (log c) t) (fma (log y) x z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -2.4e+218) {
		tmp = fma(y, i, (x * log(y)));
	} else if (x <= 1.1e+156) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = fma((b - 0.5), log(c), t) + fma(log(y), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -2.4e+218)
		tmp = fma(y, i, Float64(x * log(y)));
	elseif (x <= 1.1e+156)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), t) + fma(log(y), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -2.4e+218], N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+156], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999981e218

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 62.2

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

   10. Applied rewrites 62.2%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 83.2

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

   13. Applied rewrites 83.2%

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

   if -2.39999999999999981e218 < x < 1.10000000000000002e156

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 1.10000000000000002e156 < 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. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, t\right) + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -2.4e+218)
     t_1
     (if (<= x 5e+228) (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a 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 = fma(y, i, (x * log(y)));
	double tmp;
	if (x <= -2.4e+218) {
		tmp = t_1;
	} else if (x <= 5e+228) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.4e+218)
		tmp = t_1;
	elseif (x <= 5e+228)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+218], t$95$1, If[LessEqual[x, 5e+228], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999981e218 or 5e228 < 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. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 49.3

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

   10. Applied rewrites 49.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 81.4

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

   13. Applied rewrites 81.4%

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

   if -2.39999999999999981e218 < x < 5e228

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -210000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= i -210000000000.0)
   (+ (fma (/ z a) a a) (* i y))
   (if (<= i 3.4e+189)
     (+ (fma (- b 0.5) (log c) z) (+ a t))
     (fma y i (* x (log y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (i <= -210000000000.0) {
		tmp = fma((z / a), a, a) + (i * y);
	} else if (i <= 3.4e+189) {
		tmp = fma((b - 0.5), log(c), z) + (a + t);
	} else {
		tmp = fma(y, i, (x * log(y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (i <= -210000000000.0)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	elseif (i <= 3.4e+189)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + Float64(a + t));
	else
		tmp = fma(y, i, Float64(x * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[i, -210000000000.0], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.4e+189], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.1e11

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.2%

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

   if -2.1e11 < i < 3.39999999999999983e189

   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 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.6%

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

   if 3.39999999999999983e189 < i

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 69.5

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

   10. Applied rewrites 69.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 84.2

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

   13. Applied rewrites 84.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -210000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{t} + \frac{z}{t}, t, t\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.4e-55)
   (+ (fma (+ (/ a t) (/ z t)) t t) (* i y))
   (+ (fma (/ z a) a a) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.4e-55) {
		tmp = fma(((a / t) + (z / t)), t, t) + (i * y);
	} else {
		tmp = fma((z / a), a, a) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.4e-55)
		tmp = Float64(fma(Float64(Float64(a / t) + Float64(z / t)), t, t) + Float64(i * y));
	else
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.4e-55], N[(N[(N[(N[(a / t), $MachinePrecision] + N[(z / t), $MachinePrecision]), $MachinePrecision] * t + t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999992e-55

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 55.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 48.6%

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

   if 1.39999999999999992e-55 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 63.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{t} + \frac{z}{t}, t, t\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{z}{t}, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.4e-55) (fma y i (fma (/ z t) t t)) (+ (fma (/ z a) a a) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.4e-55) {
		tmp = fma(y, i, fma((z / t), t, t));
	} else {
		tmp = fma((z / a), a, a) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.4e-55)
		tmp = fma(y, i, fma(Float64(z / t), t, t));
	else
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.4e-55], N[(y * i + N[(N[(z / t), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999992e-55

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 55.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 55.7

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

   10. Applied rewrites 55.8%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 44.1%

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

   if 1.39999999999999992e-55 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 63.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{z}{t}, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.1% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

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 = (i * y) + (a + t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 83.9

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

5. Applied rewrites 83.9%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 47.8%

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

9. Final simplification 47.8%

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

Alternative 18: 24.2% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

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 = i * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 19.5

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

5. Applied rewrites 19.5%

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

6. Final simplification 19.5%

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

Reproduce

herbie shell --seed 2024270 
(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)))