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

Percentage Accurate: 99.8% → 99.8%

Time: 13.6s

Alternatives: 9

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: 29.3% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision], N[(i * y), $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)) < -100

   1. Initial program 99.8%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 34.5%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 28.1%

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

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

   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. lower-*.f64 23.3

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

   5. Applied rewrites 23.3%

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

4. Final simplification 31.3%

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

Alternative 3: 30.2% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision], N[(i * y), $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)) < -inf.0 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 28.1%

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

4. Final simplification 33.3%

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

Alternative 4: 31.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 34.5%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 42.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 42.2

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

   10. Applied rewrites 42.2%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 28.4%

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

4. Final simplification 31.5%

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

Alternative 5: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -9.5e+121)
     t_1
     (if (<= x 1.75e+196)
       (+ (fma (- b 0.5) (log c) (fma i y 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 <= -9.5e+121) {
		tmp = t_1;
	} else if (x <= 1.75e+196) {
		tmp = fma((b - 0.5), log(c), fma(i, y, 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 <= -9.5e+121)
		tmp = t_1;
	elseif (x <= 1.75e+196)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, 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, -9.5e+121], t$95$1, If[LessEqual[x, 1.75e+196], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999949e121 or 1.7499999999999999e196 < x

   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 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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -9.49999999999999949e121 < x < 1.7499999999999999e196

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ a z) z z)))
   (if (<= z -1.85e+170)
     (fma (/ (* i y) z) z z)
     (if (<= z -1.9e+126)
       t_1
       (if (<= z -1.8e+48)
         (fma y i (* x (log y)))
         (if (<= z -6.6e+22)
           t_1
           (if (<= z 5.6e-67) (fma y i (* b (log c))) 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((a / z), z, z);
	double tmp;
	if (z <= -1.85e+170) {
		tmp = fma(((i * y) / z), z, z);
	} else if (z <= -1.9e+126) {
		tmp = t_1;
	} else if (z <= -1.8e+48) {
		tmp = fma(y, i, (x * log(y)));
	} else if (z <= -6.6e+22) {
		tmp = t_1;
	} else if (z <= 5.6e-67) {
		tmp = fma(y, i, (b * log(c)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(a / z), z, z)
	tmp = 0.0
	if (z <= -1.85e+170)
		tmp = fma(Float64(Float64(i * y) / z), z, z);
	elseif (z <= -1.9e+126)
		tmp = t_1;
	elseif (z <= -1.8e+48)
		tmp = fma(y, i, Float64(x * log(y)));
	elseif (z <= -6.6e+22)
		tmp = t_1;
	elseif (z <= 5.6e-67)
		tmp = fma(y, i, Float64(b * log(c)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision]}, If[LessEqual[z, -1.85e+170], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], If[LessEqual[z, -1.9e+126], t$95$1, If[LessEqual[z, -1.8e+48], N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -6.6e+22], t$95$1, If[LessEqual[z, 5.6e-67], N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.84999999999999994e170

   1. Initial program 99.8%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.7%

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

   if -1.84999999999999994e170 < z < -1.90000000000000008e126 or -1.79999999999999992e48 < z < -6.5999999999999996e22 or 5.60000000000000021e-67 < z

   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 z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 39.7%

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

   if -1.90000000000000008e126 < z < -1.79999999999999992e48

   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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 43.9

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

   7. Applied rewrites 43.9%

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

   if -6.5999999999999996e22 < z < 5.60000000000000021e-67

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 45.5

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

   10. Applied rewrites 45.5%

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

   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. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 47.9

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

   13. Applied rewrites 47.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \left(\frac{a}{x} + \log y\right) \cdot x\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+234}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (+ (/ a x) (log y)) x))))
   (if (<= x -8.5e+234)
     t_1
     (if (<= x 2.3e+194)
       (+ (fma (- b 0.5) (log c) (fma i y 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, (((a / x) + log(y)) * x));
	double tmp;
	if (x <= -8.5e+234) {
		tmp = t_1;
	} else if (x <= 2.3e+194) {
		tmp = fma((b - 0.5), log(c), fma(i, y, 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(Float64(Float64(a / x) + log(y)) * x))
	tmp = 0.0
	if (x <= -8.5e+234)
		tmp = t_1;
	elseif (x <= 2.3e+194)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, 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[(N[(N[(a / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+234], t$95$1, If[LessEqual[x, 2.3e+194], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999989e234 or 2.30000000000000005e194 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 82.1

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

   10. Applied rewrites 82.1%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 83.0%

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

   if -8.49999999999999989e234 < x < 2.30000000000000005e194

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\frac{a}{x} + \log y\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\frac{a}{x} + \log y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -6.8e+246)
     t_1
     (if (<= x 2.6e+218)
       (+ (fma (- b 0.5) (log c) (fma i y 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 <= -6.8e+246) {
		tmp = t_1;
	} else if (x <= 2.6e+218) {
		tmp = fma((b - 0.5), log(c), fma(i, y, 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 <= -6.8e+246)
		tmp = t_1;
	elseif (x <= 2.6e+218)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, 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, -6.8e+246], t$95$1, If[LessEqual[x, 2.6e+218], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999977e246 or 2.60000000000000002e218 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.2

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

   5. Applied rewrites 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if -6.79999999999999977e246 < x < 2.60000000000000002e218

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.9% 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. lower-*.f64 23.3

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

5. Applied rewrites 23.3%

   \[\leadsto \color{blue}{i \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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)))