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

Percentage Accurate: 99.8% → 99.8%

Time: 14.8s

Alternatives: 17

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

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

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]

Derivation

1. Initial program 99.8%

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

Alternative 2: 37.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;t\_1 + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 -2e+66)
     (* i (+ y (/ z i)))
     (if (<= t_2 1e+306) (+ a (* b (log c))) (+ t_1 (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -2e+66) {
		tmp = i * (y + (z / i));
	} else if (t_2 <= 1e+306) {
		tmp = a + (b * log(c));
	} else {
		tmp = t_1 + (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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_2 <= (-2d+66)) then
        tmp = i * (y + (z / i))
    else if (t_2 <= 1d+306) then
        tmp = a + (b * log(c))
    else
        tmp = t_1 + (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 = x * Math.log(y);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_2 <= -2e+66) {
		tmp = i * (y + (z / i));
	} else if (t_2 <= 1e+306) {
		tmp = a + (b * Math.log(c));
	} else {
		tmp = t_1 + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x * math.log(y)
	t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_2 <= -2e+66:
		tmp = i * (y + (z / i))
	elif t_2 <= 1e+306:
		tmp = a + (b * math.log(c))
	else:
		tmp = t_1 + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -2e+66)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	elseif (t_2 <= 1e+306)
		tmp = Float64(a + Float64(b * log(c)));
	else
		tmp = Float64(t_1 + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x * log(y);
	t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_2 <= -2e+66)
		tmp = i * (y + (z / i));
	elseif (t_2 <= 1e+306)
		tmp = a + (b * log(c));
	else
		tmp = t_1 + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+66], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+306], N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.99999999999999989e66

   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 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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 74.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 33.0

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

   8. Simplified 33.0%

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

   if -1.99999999999999989e66 < (+.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)) < 1.00000000000000002e306

   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. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 28.3

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

   8. Simplified 28.3%

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

   if 1.00000000000000002e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 94.5

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

   5. Simplified 94.5%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+306}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+306}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+66)
     (* i (+ y (/ z i)))
     (if (<= t_1 1e+306) (+ a (* b (log c))) (fma (log c) b (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = i * (y + (z / i));
	} else if (t_1 <= 1e+306) {
		tmp = a + (b * log(c));
	} else {
		tmp = fma(log(c), b, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+66)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	elseif (t_1 <= 1e+306)
		tmp = Float64(a + Float64(b * log(c)));
	else
		tmp = fma(log(c), b, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+66], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+306], N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[c], $MachinePrecision] * b + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.99999999999999989e66

   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 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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 74.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 33.0

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

   8. Simplified 33.0%

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

   if -1.99999999999999989e66 < (+.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)) < 1.00000000000000002e306

   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. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 28.3

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

   8. Simplified 28.3%

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

   if 1.00000000000000002e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.4

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

   5. Simplified 94.4%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 94.4

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

   7. Applied egg-rr 94.4%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+306}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+66)
     (* i (+ y (/ z i)))
     (if (<= t_1 1e+299) (+ a (* b (log c))) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = i * (y + (z / i));
	} else if (t_1 <= 1e+299) {
		tmp = a + (b * log(c));
	} else {
		tmp = a + (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 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-2d+66)) then
        tmp = i * (y + (z / i))
    else if (t_1 <= 1d+299) then
        tmp = a + (b * log(c))
    else
        tmp = a + (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 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = i * (y + (z / i));
	} else if (t_1 <= 1e+299) {
		tmp = a + (b * Math.log(c));
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -2e+66:
		tmp = i * (y + (z / i))
	elif t_1 <= 1e+299:
		tmp = a + (b * math.log(c))
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+66)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	elseif (t_1 <= 1e+299)
		tmp = Float64(a + Float64(b * log(c)));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -2e+66)
		tmp = i * (y + (z / i));
	elseif (t_1 <= 1e+299)
		tmp = a + (b * log(c));
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+66], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.99999999999999989e66

   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 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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 74.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 33.0

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

   8. Simplified 33.0%

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

   if -1.99999999999999989e66 < (+.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)) < 1.0000000000000001e299

   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. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 28.1

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

   8. Simplified 28.1%

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

   if 1.0000000000000001e299 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 95.2

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

   5. Simplified 95.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 80.8

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

   8. Simplified 80.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+299}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+305)
     (* y i)
     (if (<= t_1 -50.0) (* i (/ z i)) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+305) {
		tmp = y * i;
	} else if (t_1 <= -50.0) {
		tmp = i * (z / i);
	} else {
		tmp = a + (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 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-5d+305)) then
        tmp = y * i
    else if (t_1 <= (-50.0d0)) then
        tmp = i * (z / i)
    else
        tmp = a + (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 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+305) {
		tmp = y * i;
	} else if (t_1 <= -50.0) {
		tmp = i * (z / i);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -5e+305:
		tmp = y * i
	elif t_1 <= -50.0:
		tmp = i * (z / i)
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+305)
		tmp = Float64(y * i);
	elseif (t_1 <= -50.0)
		tmp = Float64(i * Float64(z / i));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -5e+305)
		tmp = y * i;
	elseif (t_1 <= -50.0)
		tmp = i * (z / i);
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+305], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -50.0], N[(i * N[(z / i), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.00000000000000009e305

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 87.2

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

   5. Simplified 87.2%

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

   if -5.00000000000000009e305 < (+.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)) < -50

   1. Initial program 99.7%

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

   3. Taylor expanded in 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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 71.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 14.0

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

   8. Simplified 14.0%

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

   if -50 < (+.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 x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 31.6

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

   8. Simplified 31.6%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+305}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{+306}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -50.0) (* y i) (if (<= t_1 1e+306) a (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -50.0) {
		tmp = y * i;
	} else if (t_1 <= 1e+306) {
		tmp = a;
	} else {
		tmp = 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 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-50.0d0)) then
        tmp = y * i
    else if (t_1 <= 1d+306) then
        tmp = a
    else
        tmp = 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 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -50.0) {
		tmp = y * i;
	} else if (t_1 <= 1e+306) {
		tmp = a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -50.0:
		tmp = y * i
	elif t_1 <= 1e+306:
		tmp = a
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = Float64(y * i);
	elseif (t_1 <= 1e+306)
		tmp = a;
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -50.0)
		tmp = y * i;
	elseif (t_1 <= 1e+306)
		tmp = a;
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 1e+306], a, N[(y * 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)) < -50 or 1.00000000000000002e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 29.6

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

   5. Simplified 29.6%

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

   if -50 < (+.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)) < 1.00000000000000002e306

   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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 62.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 8.7

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

   8. Simplified 8.7%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{1}{i}\right)} \cdot i \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{i} \cdot i\right)} \]

      6. inv-pow N/A

         \[\leadsto a \cdot \left(\color{blue}{{i}^{-1}} \cdot i\right) \]

      7. pow-plus N/A

         \[\leadsto a \cdot \color{blue}{{i}^{\left(-1 + 1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto a \cdot {i}^{\color{blue}{0}} \]

      9. metadata-eval N/A

         \[\leadsto a \cdot \color{blue}{1} \]

      10. metadata-eval N/A

          \[\leadsto a \cdot \color{blue}{\left(-1 \cdot -1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot -1\right)} \]

      12. metadata-eval 14.9

          \[\leadsto a \cdot \color{blue}{1} \]

   10. Applied egg-rr 14.9%

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

       1. *-rgt-identity 14.9

          \[\leadsto \color{blue}{a} \]

   12. Applied egg-rr 14.9%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+306}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   (* i (+ y (/ z i)))
   (+ a (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = a + (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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-50.0d0)) then
        tmp = i * (y + (z / i))
    else
        tmp = a + (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 (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -50.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -50.0:
		tmp = i * (y + (z / i))
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -50.0)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0)
		tmp = i * (y + (z / i));
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   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 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. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 74.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 31.6

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

   8. Simplified 31.6%

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

   if -50 < (+.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 x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 31.6

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

   8. Simplified 31.6%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+177}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ z (fma x (log y) (fma (log c) (+ b -0.5) a)))))
   (if (<= x -5.6e+135)
     t_1
     (if (<= x 5.4e+177)
       (+ (* y i) (+ t (+ a (fma (log c) (+ b -0.5) z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = z + fma(x, log(y), fma(log(c), (b + -0.5), a));
	double tmp;
	if (x <= -5.6e+135) {
		tmp = t_1;
	} else if (x <= 5.4e+177) {
		tmp = (y * i) + (t + (a + fma(log(c), (b + -0.5), z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(z + fma(x, log(y), fma(log(c), Float64(b + -0.5), a)))
	tmp = 0.0
	if (x <= -5.6e+135)
		tmp = t_1;
	elseif (x <= 5.4e+177)
		tmp = Float64(Float64(y * i) + Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+135], t$95$1, If[LessEqual[x, 5.4e+177], N[(N[(y * i), $MachinePrecision] + N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.60000000000000004e135 or 5.39999999999999982e177 < 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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 93.2

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

   5. Simplified 93.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 82.0

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

   8. Simplified 82.0%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-log.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. sub-neg N/A

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

       14. metadata-eval N/A

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

       15. lower-+.f64 83.4

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

   11. Simplified 83.4%

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

   if -5.60000000000000004e135 < x < 5.39999999999999982e177

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 95.8

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

   5. Simplified 95.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+135}:\\ \;\;\;\;z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+177}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \mathbf{if}\;x \leq -1.58 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x (log y) (fma (log c) (+ b -0.5) a))))
   (if (<= x -1.58e+165)
     t_1
     (if (<= x 2e+181)
       (+ (* y i) (+ t (+ a (fma (log c) (+ b -0.5) z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, log(y), fma(log(c), (b + -0.5), a));
	double tmp;
	if (x <= -1.58e+165) {
		tmp = t_1;
	} else if (x <= 2e+181) {
		tmp = (y * i) + (t + (a + fma(log(c), (b + -0.5), z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, log(y), fma(log(c), Float64(b + -0.5), a))
	tmp = 0.0
	if (x <= -1.58e+165)
		tmp = t_1;
	elseif (x <= 2e+181)
		tmp = Float64(Float64(y * i) + Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.58e+165], t$95$1, If[LessEqual[x, 2e+181], N[(N[(y * i), $MachinePrecision] + N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.58000000000000007e165 or 1.9999999999999998e181 < 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. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 53.0

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

   5. Simplified 53.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 45.9

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

   8. Simplified 45.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-log.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. lower-+.f64 75.5

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

   11. Simplified 75.5%

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

   if -1.58000000000000007e165 < x < 1.9999999999999998e181

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 95.4

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

   5. Simplified 95.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.58 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 87.1

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

5. Simplified 87.1%

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

6. Final simplification 87.1%

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

Alternative 11: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y + y \cdot i\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+227}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x (log y)) (* y i))))
   (if (<= x -2.95e+169)
     t_1
     (if (<= x 6e+227)
       (+ (* y i) (+ t (+ a (fma (log c) (+ b -0.5) z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * log(y)) + (y * i);
	double tmp;
	if (x <= -2.95e+169) {
		tmp = t_1;
	} else if (x <= 6e+227) {
		tmp = (y * i) + (t + (a + fma(log(c), (b + -0.5), z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * log(y)) + Float64(y * i))
	tmp = 0.0
	if (x <= -2.95e+169)
		tmp = t_1;
	elseif (x <= 6e+227)
		tmp = Float64(Float64(y * i) + Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.95e+169], t$95$1, If[LessEqual[x, 6e+227], N[(N[(y * i), $MachinePrecision] + N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.95e169 or 5.99999999999999972e227 < 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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 77.1

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

   5. Simplified 77.1%

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

   if -2.95e169 < x < 5.99999999999999972e227

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 94.1

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

   5. Simplified 94.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+227}:\\ \;\;\;\;y \cdot i + \left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y + y \cdot i\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+227}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x (log y)) (* y i))))
   (if (<= x -2.95e+169)
     t_1
     (if (<= x 6e+227) (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * log(y)) + (y * i);
	double tmp;
	if (x <= -2.95e+169) {
		tmp = t_1;
	} else if (x <= 6e+227) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * log(y)) + Float64(y * i))
	tmp = 0.0
	if (x <= -2.95e+169)
		tmp = t_1;
	elseif (x <= 6e+227)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.95e+169], t$95$1, If[LessEqual[x, 6e+227], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.95e169 or 5.99999999999999972e227 < 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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 77.1

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

   5. Simplified 77.1%

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

   if -2.95e169 < x < 5.99999999999999972e227

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 94.1

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

   5. Simplified 94.1%

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

Alternative 13: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y + y \cdot i\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+227}:\\ \;\;\;\;a + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x (log y)) (* y i))))
   (if (<= x -2.95e+169)
     t_1
     (if (<= x 6e+227) (+ a (fma (log c) (+ b -0.5) (fma i y z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * log(y)) + (y * i);
	double tmp;
	if (x <= -2.95e+169) {
		tmp = t_1;
	} else if (x <= 6e+227) {
		tmp = a + fma(log(c), (b + -0.5), fma(i, y, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * log(y)) + Float64(y * i))
	tmp = 0.0
	if (x <= -2.95e+169)
		tmp = t_1;
	elseif (x <= 6e+227)
		tmp = Float64(a + fma(log(c), Float64(b + -0.5), fma(i, y, z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.95e+169], t$95$1, If[LessEqual[x, 6e+227], N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.95e169 or 5.99999999999999972e227 < 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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 77.1

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

   5. Simplified 77.1%

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

   if -2.95e169 < x < 5.99999999999999972e227

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 85.1

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

   5. Simplified 85.1%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 71.6

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

   8. Simplified 71.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. associate-+r+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-log.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 79.4

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

   11. Simplified 79.4%

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

Alternative 14: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y + y \cdot i\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+92}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x (log y)) (* y i))))
   (if (<= x -4.3e+163)
     t_1
     (if (<= x 3.5e+92) (+ a (+ t (fma (log c) (+ b -0.5) z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * log(y)) + (y * i);
	double tmp;
	if (x <= -4.3e+163) {
		tmp = t_1;
	} else if (x <= 3.5e+92) {
		tmp = a + (t + fma(log(c), (b + -0.5), z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * log(y)) + Float64(y * i))
	tmp = 0.0
	if (x <= -4.3e+163)
		tmp = t_1;
	elseif (x <= 3.5e+92)
		tmp = Float64(a + Float64(t + fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e+163], t$95$1, If[LessEqual[x, 3.5e+92], N[(a + N[(t + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3000000000000002e163 or 3.49999999999999986e92 < 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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 69.9

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

   5. Simplified 69.9%

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

   if -4.3000000000000002e163 < x < 3.49999999999999986e92

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 78.1

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

   8. Simplified 78.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+92}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(i, y, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -2.3e+122)
   (+ a (+ t (fma (log c) (+ b -0.5) z)))
   (fma (log c) (+ b -0.5) (fma i y a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2.3e+122) {
		tmp = a + (t + fma(log(c), (b + -0.5), z));
	} else {
		tmp = fma(log(c), (b + -0.5), fma(i, y, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2.3e+122)
		tmp = Float64(a + Float64(t + fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = fma(log(c), Float64(b + -0.5), fma(i, y, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2.3e+122], N[(a + N[(t + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + N[(i * y + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e122

   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. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 73.5

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

   8. Simplified 73.5%

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

   if -2.3000000000000001e122 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 60.2

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

   8. Simplified 60.2%

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

   9. Taylor expanded in x around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 53.9

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

   11. Simplified 53.9%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(i, y, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 82.3

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

5. Simplified 82.3%

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

6. Taylor expanded in i around inf

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

   1. lower-*.f64 34.8

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

8. Simplified 34.8%

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

9. Final simplification 34.8%

   \[\leadsto a + y \cdot i \]
10. Add Preprocessing

Alternative 17: 16.5% accurate, 234.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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. distribute-rgt-neg-in N/A

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

   3. distribute-lft-out N/A

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

5. Simplified 70.2%

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

6. Taylor expanded in a around inf

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

   1. lower-/.f64 10.4

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

8. Simplified 10.4%

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

   1. lift-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

      \[\leadsto \color{blue}{\left(a \cdot \frac{1}{i}\right)} \cdot i \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{1}{i} \cdot i\right)} \]

   6. inv-pow N/A

      \[\leadsto a \cdot \left(\color{blue}{{i}^{-1}} \cdot i\right) \]

   7. pow-plus N/A

      \[\leadsto a \cdot \color{blue}{{i}^{\left(-1 + 1\right)}} \]

   8. metadata-eval N/A

      \[\leadsto a \cdot {i}^{\color{blue}{0}} \]

   9. metadata-eval N/A

      \[\leadsto a \cdot \color{blue}{1} \]

   10. metadata-eval N/A

       \[\leadsto a \cdot \color{blue}{\left(-1 \cdot -1\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{a \cdot \left(-1 \cdot -1\right)} \]

   12. metadata-eval 16.2

       \[\leadsto a \cdot \color{blue}{1} \]

10. Applied egg-rr 16.2%

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

    1. *-rgt-identity 16.2

       \[\leadsto \color{blue}{a} \]

12. Applied egg-rr 16.2%

    \[\leadsto \color{blue}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))