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

Percentage Accurate: 99.8% → 99.8%

Time: 13.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.9

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

   16. lift-+.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 27.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot y + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+258}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{i} + y\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* i y)
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))))))
   (if (<= t_1 -5e+258)
     (* i y)
     (if (<= t_1 -5e+94) (* (/ z i) i) (* (+ (/ a i) y) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c)));
	double tmp;
	if (t_1 <= -5e+258) {
		tmp = i * y;
	} else if (t_1 <= -5e+94) {
		tmp = (z / i) * i;
	} else {
		tmp = ((a / i) + y) * i;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c)))
    if (t_1 <= (-5d+258)) then
        tmp = i * y
    else if (t_1 <= (-5d+94)) then
        tmp = (z / i) * i
    else
        tmp = ((a / i) + y) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c)));
	double tmp;
	if (t_1 <= -5e+258) {
		tmp = i * y;
	} else if (t_1 <= -5e+94) {
		tmp = (z / i) * i;
	} else {
		tmp = ((a / i) + y) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (i * y) + (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c)))
	tmp = 0
	if t_1 <= -5e+258:
		tmp = i * y
	elif t_1 <= -5e+94:
		tmp = (z / i) * i
	else:
		tmp = ((a / i) + y) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(i * y) + Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))))
	tmp = 0.0
	if (t_1 <= -5e+258)
		tmp = Float64(i * y);
	elseif (t_1 <= -5e+94)
		tmp = Float64(Float64(z / i) * i);
	else
		tmp = Float64(Float64(Float64(a / i) + y) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c)));
	tmp = 0.0;
	if (t_1 <= -5e+258)
		tmp = i * y;
	elseif (t_1 <= -5e+94)
		tmp = (z / i) * i;
	else
		tmp = ((a / i) + y) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y), $MachinePrecision] + 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]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+258], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -5e+94], N[(N[(z / i), $MachinePrecision] * i), $MachinePrecision], N[(N[(N[(a / i), $MachinePrecision] + y), $MachinePrecision] * i), $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)) < -5e258

   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 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 45.3

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

   5. Applied rewrites 45.3%

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

   if -5e258 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000001e94

   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 i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{z}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 10.6%

      \[\leadsto \frac{z}{i} \cdot i \]

   if -5.0000000000000001e94 < (+.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 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 33.6%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) \leq -5 \cdot 10^{+258}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{i} + y\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, b \cdot \log c\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* b (log c)))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -1e+147)
     t_1
     (if (<= t_2 2e+167) (fma y i (fma (log c) -0.5 (+ (+ z t) a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (b * log(c)));
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -1e+147) {
		tmp = t_1;
	} else if (t_2 <= 2e+167) {
		tmp = fma(y, i, fma(log(c), -0.5, ((z + t) + a)));
	} 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 <= -1e+147)
		tmp = t_1;
	elseif (t_2 <= 2e+167)
		tmp = fma(y, i, fma(log(c), -0.5, Float64(Float64(z + t) + a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(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, -1e+147], t$95$1, If[LessEqual[t$95$2, 2e+167], N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $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)) < -9.9999999999999998e146 or 2.0000000000000001e167 < (*.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 b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 76.0

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

   5. Applied rewrites 76.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if -9.9999999999999998e146 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 2.0000000000000001e167

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 79.0%

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

Alternative 4: 30.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* i y) (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))))
      -100.0)
   (fma y i (* (/ z x) x))
   (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 (((i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c)))) <= -100.0) {
		tmp = fma(y, i, ((z / x) * x));
	} 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(i * y) + Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c)))) <= -100.0)
		tmp = fma(y, i, Float64(Float64(z / x) * x));
	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[(i * y), $MachinePrecision] + 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]), $MachinePrecision], -100.0], N[(y * i + N[(N[(z / x), $MachinePrecision] * x), $MachinePrecision]), $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 99.9%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 36.2%

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

   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 99.9%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 32.2%

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

4. Final simplification 34.2%

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

Alternative 5: 31.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* i y) (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))))
      -100.0)
   (* (+ (/ z i) y) i)
   (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 (((i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c)))) <= -100.0) {
		tmp = ((z / i) + y) * i;
	} 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(i * y) + Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c)))) <= -100.0)
		tmp = Float64(Float64(Float64(z / i) + y) * i);
	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[(i * y), $MachinePrecision] + 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]), $MachinePrecision], -100.0], N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $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 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 33.7%

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

   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 99.9%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 32.2%

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

4. Final simplification 32.9%

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

Alternative 6: 31.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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
    real(8) :: tmp
    if (((i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c)))) <= (-100.0d0)) then
        tmp = ((z / i) + y) * i
    else
        tmp = ((a / i) + y) * i
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((i * y) + (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c)))) <= -100.0:
		tmp = ((z / i) + y) * i
	else:
		tmp = ((a / i) + y) * i
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((i * y) + (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c)))) <= -100.0)
		tmp = ((z / i) + y) * i;
	else
		tmp = ((a / i) + y) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(i * y), $MachinePrecision] + 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]), $MachinePrecision], -100.0], N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision], N[(N[(N[(a / i), $MachinePrecision] + y), $MachinePrecision] * i), $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 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 33.7%

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

   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 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 34.7%

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

4. Final simplification 34.2%

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

Alternative 7: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 3.9e-84)
   (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ t a))
   (fma y i (fma (log c) (- b 0.5) (+ (+ z t) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 3.9e-84) {
		tmp = fma((b - 0.5), log(c), fma(log(y), x, z)) + (t + a);
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((z + t) + a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 3.9e-84)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, z)) + Float64(t + a));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(z + t) + a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 3.9e-84], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.90000000000000023e-84

   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 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 97.9

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

   5. Applied rewrites 97.9%

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

   if 3.90000000000000023e-84 < y

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 92.4

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

   7. Applied rewrites 92.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\frac{z}{x} + \log y\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log y, x, \frac{a}{x} \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -3.7e+243)
   (fma y i (* (+ (/ z x) (log y)) x))
   (if (<= x 6e+223)
     (fma y i (fma (log c) (- b 0.5) (+ (+ z t) a)))
     (fma y i (fma (log y) x (* (/ 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 (x <= -3.7e+243) {
		tmp = fma(y, i, (((z / x) + log(y)) * x));
	} else if (x <= 6e+223) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((z + t) + a)));
	} else {
		tmp = fma(y, i, fma(log(y), x, ((a / x) * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -3.7e+243)
		tmp = fma(y, i, Float64(Float64(Float64(z / x) + log(y)) * x));
	elseif (x <= 6e+223)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(z + t) + a)));
	else
		tmp = fma(y, i, fma(log(y), x, Float64(Float64(a / x) * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -3.7e+243], N[(y * i + N[(N[(N[(z / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e+223], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[y], $MachinePrecision] * x + N[(N[(a / x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e243

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 75.0%

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

   if -3.7000000000000002e243 < x < 6.00000000000000002e223

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if 6.00000000000000002e223 < x

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 83.2%

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

   12. Applied rewrites 83.3%

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

Alternative 9: 90.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\frac{z}{x} + \log y\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\frac{a}{x} + \log y\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -3.7e+243)
   (fma y i (* (+ (/ z x) (log y)) x))
   (if (<= x 6e+223)
     (fma y i (fma (log c) (- b 0.5) (+ (+ z t) a)))
     (fma y i (* (+ (/ a x) (log y)) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -3.7e+243) {
		tmp = fma(y, i, (((z / x) + log(y)) * x));
	} else if (x <= 6e+223) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((z + t) + a)));
	} else {
		tmp = fma(y, i, (((a / x) + log(y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -3.7e+243)
		tmp = fma(y, i, Float64(Float64(Float64(z / x) + log(y)) * x));
	elseif (x <= 6e+223)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(z + t) + a)));
	else
		tmp = fma(y, i, Float64(Float64(Float64(a / x) + log(y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -3.7e+243], N[(y * i + N[(N[(N[(z / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e+223], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(N[(a / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e243

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 75.0%

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

   if -3.7000000000000002e243 < x < 6.00000000000000002e223

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if 6.00000000000000002e223 < x

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 83.2%

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

Alternative 10: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \left(\frac{a}{x} + \log y\right) \cdot x\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (+ (/ a x) (log y)) x))))
   (if (<= x -1.02e+119)
     t_1
     (if (<= x 6e+223) (fma y i (fma (log c) (- b 0.5) (+ (+ z t) a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (((a / x) + log(y)) * x));
	double tmp;
	if (x <= -1.02e+119) {
		tmp = t_1;
	} else if (x <= 6e+223) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((z + t) + a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(Float64(Float64(a / x) + log(y)) * x))
	tmp = 0.0
	if (x <= -1.02e+119)
		tmp = t_1;
	elseif (x <= 6e+223)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(z + t) + a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[(N[(a / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.02e+119], t$95$1, If[LessEqual[x, 6e+223], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e119 or 6.00000000000000002e223 < x

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 67.3%

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

   if -1.02e119 < x < 6.00000000000000002e223

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 11: 85.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 9.5e+272)
   (fma y i (fma (log c) (- b 0.5) (+ (+ z t) a)))
   (* 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 (x <= 9.5e+272) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((z + t) + a)));
	} else {
		tmp = x * log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 9.5e+272)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(z + t) + a)));
	else
		tmp = Float64(x * log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 9.5e+272], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5e272

   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. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 88.8

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

   7. Applied rewrites 88.8%

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

   if 9.5e272 < 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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 62.5

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

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(z + t\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 9.5e+272)
   (+ (fma (- b 0.5) (log c) (fma y i z)) (+ t a))
   (* 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 (x <= 9.5e+272) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (t + a);
	} else {
		tmp = x * log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 9.5e+272)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(t + a));
	else
		tmp = Float64(x * log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 9.5e+272], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5e272

   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 88.8

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

   5. Applied rewrites 88.8%

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

   if 9.5e272 < 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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 62.5

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

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 4.8e-208) (* (/ z i) i) (if (<= y 1.8e+46) (* (/ a i) i) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 4.8e-208) {
		tmp = (z / i) * i;
	} else if (y <= 1.8e+46) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 4.8d-208) then
        tmp = (z / i) * i
    else if (y <= 1.8d+46) then
        tmp = (a / i) * i
    else
        tmp = i * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 4.8e-208) {
		tmp = (z / i) * i;
	} else if (y <= 1.8e+46) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 4.8e-208:
		tmp = (z / i) * i
	elif y <= 1.8e+46:
		tmp = (a / i) * i
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 4.8e-208)
		tmp = Float64(Float64(z / i) * i);
	elseif (y <= 1.8e+46)
		tmp = Float64(Float64(a / i) * i);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= 4.8e-208)
		tmp = (z / i) * i;
	elseif (y <= 1.8e+46)
		tmp = (a / i) * i;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 4.8e-208], N[(N[(z / i), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[y, 1.8e+46], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision], N[(i * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.7999999999999998e-208

   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 i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{z}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 9.2%

      \[\leadsto \frac{z}{i} \cdot i \]

   if 4.7999999999999998e-208 < y < 1.7999999999999999e46

   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 i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 15.8%

      \[\leadsto \frac{a}{i} \cdot i \]

   if 1.7999999999999999e46 < y

   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 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 53.9

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

   5. Applied rewrites 53.9%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.8e+46) (* (/ a i) i) (* i y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.8e+46) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 1.8d+46) then
        tmp = (a / i) * i
    else
        tmp = i * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.8e+46) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 1.8e+46:
		tmp = (a / i) * i
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.8e+46)
		tmp = Float64(Float64(a / i) * i);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= 1.8e+46)
		tmp = (a / i) * i;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.8e+46], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision], N[(i * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7999999999999999e46

   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 i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 14.9%

      \[\leadsto \frac{a}{i} \cdot i \]

   if 1.7999999999999999e46 < y

   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 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 53.9

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{y \cdot i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.8% 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 99.9%

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

3. Taylor expanded in 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 27.5

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

5. Applied rewrites 27.5%

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

6. Final simplification 27.5%

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

Reproduce

herbie shell --seed 2024268 
(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)))