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

Percentage Accurate: 99.8% → 99.8%

Time: 13.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 47.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ t_2 := \log c \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+225}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, i, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_2 + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \frac{\left(t - a\right) \cdot \left(a + t\right)}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x)))))))
        (t_2 (* (log c) b)))
   (if (<= t_1 -1e+306)
     (* i y)
     (if (<= t_1 -1e+225)
       (+ (* (/ z y) y) (+ a t))
       (if (<= t_1 5e+15)
         (fma y i t_2)
         (if (<= t_1 2e+302)
           (+ t_2 (+ a t))
           (+ (* i y) (/ (* (- t a) (+ a t)) (- t a)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double t_2 = log(c) * b;
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = i * y;
	} else if (t_1 <= -1e+225) {
		tmp = ((z / y) * y) + (a + t);
	} else if (t_1 <= 5e+15) {
		tmp = fma(y, i, t_2);
	} else if (t_1 <= 2e+302) {
		tmp = t_2 + (a + t);
	} else {
		tmp = (i * y) + (((t - a) * (a + t)) / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(i * y) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x))))))
	t_2 = Float64(log(c) * b)
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = Float64(i * y);
	elseif (t_1 <= -1e+225)
		tmp = Float64(Float64(Float64(z / y) * y) + Float64(a + t));
	elseif (t_1 <= 5e+15)
		tmp = fma(y, i, t_2);
	elseif (t_1 <= 2e+302)
		tmp = Float64(t_2 + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(Float64(Float64(t - a) * Float64(a + t)) / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+306], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -1e+225], N[(N[(N[(z / y), $MachinePrecision] * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+15], N[(y * i + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(t$95$2 + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(N[(N[(t - a), $MachinePrecision] * N[(a + t), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 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)) < -1.00000000000000002e306

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000002e306 < (+.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)) < -9.99999999999999928e224

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.2%

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

   if -9.99999999999999928e224 < (+.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)) < 5e15

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 42.8

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

   5. Applied rewrites 42.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 42.8

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

   7. Applied rewrites 42.8%

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

   if 5e15 < (+.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)) < 2.0000000000000002e302

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 53.9%

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

   if 2.0000000000000002e302 < (+.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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.3%

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

   10. Applied rewrites 74.4%

       \[\leadsto \frac{\left(t + a\right) \cdot \left(t - a\right)}{t - a} + \color{blue}{y} \cdot i \]
3. Recombined 5 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+225}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log c \cdot b\right)\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \frac{\left(t - a\right) \cdot \left(a + t\right)}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + t\_1\right)\right)\right)\right)\\ t_3 := t\_1 + i \cdot y\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z t_1))))))
        (t_3 (+ t_1 (* i y))))
   (if (<= t_2 -4e+303)
     t_3
     (if (<= t_2 100.0)
       (+ (fma -0.5 (log c) z) (+ a t))
       (if (<= t_2 2e+302) (+ (* (log c) b) (+ a t)) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	double t_3 = t_1 + (i * y);
	double tmp;
	if (t_2 <= -4e+303) {
		tmp = t_3;
	} else if (t_2 <= 100.0) {
		tmp = fma(-0.5, log(c), z) + (a + t);
	} else if (t_2 <= 2e+302) {
		tmp = (log(c) * b) + (a + t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(i * y) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + t_1)))))
	t_3 = Float64(t_1 + Float64(i * y))
	tmp = 0.0
	if (t_2 <= -4e+303)
		tmp = t_3;
	elseif (t_2 <= 100.0)
		tmp = Float64(fma(-0.5, log(c), z) + Float64(a + t));
	elseif (t_2 <= 2e+302)
		tmp = Float64(Float64(log(c) * b) + Float64(a + t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+303], t$95$3, If[LessEqual[t$95$2, 100.0], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+302], N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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)) < -4e303 or 2.0000000000000002e302 < (+.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.4%

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

   if 100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.0000000000000002e302

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 53.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \frac{\left(t - a\right) \cdot \left(a + t\right)}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* i y)
          (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))))
   (if (<= t_1 -1e+306)
     (* i y)
     (if (<= t_1 100.0)
       (+ (fma -0.5 (log c) z) (+ a t))
       (if (<= t_1 2e+302)
         (+ (* (log c) b) (+ a t))
         (+ (* i y) (/ (* (- t a) (+ a t)) (- t a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = i * y;
	} else if (t_1 <= 100.0) {
		tmp = fma(-0.5, log(c), z) + (a + t);
	} else if (t_1 <= 2e+302) {
		tmp = (log(c) * b) + (a + t);
	} else {
		tmp = (i * y) + (((t - a) * (a + t)) / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(i * y) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x))))))
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = Float64(i * y);
	elseif (t_1 <= 100.0)
		tmp = Float64(fma(-0.5, log(c), z) + Float64(a + t));
	elseif (t_1 <= 2e+302)
		tmp = Float64(Float64(log(c) * b) + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(Float64(Float64(t - a) * Float64(a + t)) / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+306], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(N[(N[(t - a), $MachinePrecision] * N[(a + t), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000002e306 < (+.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)) < 100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 53.5%

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

   if 100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.0000000000000002e302

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 53.3%

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

   if 2.0000000000000002e302 < (+.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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.3%

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

   10. Applied rewrites 74.4%

       \[\leadsto \frac{\left(t + a\right) \cdot \left(t - a\right)}{t - a} + \color{blue}{y} \cdot i \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \frac{\left(t - a\right) \cdot \left(a + t\right)}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + t\_1\right)\right)\right)\right)\\ t_3 := t\_1 + i \cdot y\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z t_1))))))
        (t_3 (+ t_1 (* i y))))
   (if (<= t_2 -4e+303)
     t_3
     (if (<= t_2 2e+302) (+ (fma (- b 0.5) (log c) z) (+ a t)) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	double t_3 = t_1 + (i * y);
	double tmp;
	if (t_2 <= -4e+303) {
		tmp = t_3;
	} else if (t_2 <= 2e+302) {
		tmp = fma((b - 0.5), log(c), z) + (a + t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(i * y) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + t_1)))))
	t_3 = Float64(t_1 + Float64(i * y))
	tmp = 0.0
	if (t_2 <= -4e+303)
		tmp = t_3;
	elseif (t_2 <= 2e+302)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + Float64(a + t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+303], t$95$3, If[LessEqual[t$95$2, 2e+302], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

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)) < -4e303 or 2.0000000000000002e302 < (+.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4e303 < (+.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)) < 2.0000000000000002e302

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ a t))))
   (if (<= x -2.05e+226)
     t_1
     (if (<= x 2.4e+151)
       (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma((b - 0.5), log(c), fma(log(y), x, z)) + (a + t);
	double tmp;
	if (x <= -2.05e+226) {
		tmp = t_1;
	} else if (x <= 2.4e+151) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, z)) + Float64(a + t))
	tmp = 0.0
	if (x <= -2.05e+226)
		tmp = t_1;
	elseif (x <= 2.4e+151)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e+226], t$95$1, If[LessEqual[x, 2.4e+151], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.04999999999999993e226 or 2.4000000000000001e151 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -2.04999999999999993e226 < x < 2.4000000000000001e151

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+243}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -4.4e+243)
   (+ (* (log y) x) (* i y))
   (if (<= x 8e+222)
     (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
     (+ (fma (- b 0.5) (log c) (fma (log y) x t)) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -4.4e+243) {
		tmp = (log(y) * x) + (i * y);
	} else if (x <= 8e+222) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = fma((b - 0.5), log(c), fma(log(y), x, t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -4.4e+243)
		tmp = Float64(Float64(log(y) * x) + Float64(i * y));
	elseif (x <= 8e+222)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -4.4e+243], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+222], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000018e243

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -4.40000000000000018e243 < x < 8.0000000000000004e222

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 8.0000000000000004e222 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 94.9

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+243}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x + i \cdot y\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+243}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log y) x) (* i y))))
   (if (<= x -4.4e+243)
     t_1
     (if (<= x 1.35e+237)
       (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (log(y) * x) + (i * y);
	double tmp;
	if (x <= -4.4e+243) {
		tmp = t_1;
	} else if (x <= 1.35e+237) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(log(y) * x) + Float64(i * y))
	tmp = 0.0
	if (x <= -4.4e+243)
		tmp = t_1;
	elseif (x <= 1.35e+237)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e+243], t$95$1, If[LessEqual[x, 1.35e+237], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000018e243 or 1.35e237 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -4.40000000000000018e243 < x < 1.35e237

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+243}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.3e+33)
   (+ (fma (- b 0.5) (log c) z) (+ a t))
   (+ (fma -0.5 (log c) (fma y i z)) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.3e+33) {
		tmp = fma((b - 0.5), log(c), z) + (a + t);
	} else {
		tmp = fma(-0.5, log(c), fma(y, i, z)) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.3e+33)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + Float64(a + t));
	else
		tmp = Float64(fma(-0.5, log(c), fma(y, i, z)) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.3e+33], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2999999999999999e33

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.9%

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

   if 1.2999999999999999e33 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-251}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-227}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1e-251)
   (+ (* (log c) b) (+ a t))
   (if (<= y 3.2e-227)
     (* (log y) x)
     (if (<= y 7.8e-59)
       (fma (/ z a) a a)
       (if (<= y 7.5e+30) (+ (* (/ z y) y) (+ a t)) (+ (* i y) (+ a t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1e-251) {
		tmp = (log(c) * b) + (a + t);
	} else if (y <= 3.2e-227) {
		tmp = log(y) * x;
	} else if (y <= 7.8e-59) {
		tmp = fma((z / a), a, a);
	} else if (y <= 7.5e+30) {
		tmp = ((z / y) * y) + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1e-251)
		tmp = Float64(Float64(log(c) * b) + Float64(a + t));
	elseif (y <= 3.2e-227)
		tmp = Float64(log(y) * x);
	elseif (y <= 7.8e-59)
		tmp = fma(Float64(z / a), a, a);
	elseif (y <= 7.5e+30)
		tmp = Float64(Float64(Float64(z / y) * y) + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1e-251], N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-227], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 7.8e-59], N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision], If[LessEqual[y, 7.5e+30], N[(N[(N[(z / y), $MachinePrecision] * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.00000000000000002e-251

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 63.8%

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

   if 1.00000000000000002e-251 < y < 3.2000000000000001e-227

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 3.2000000000000001e-227 < y < 7.80000000000000038e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 35.0%

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

   if 7.80000000000000038e-59 < y < 7.49999999999999973e30

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 55.3%

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

   if 7.49999999999999973e30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 68.7%

      \[\leadsto \left(t + a\right) + y \cdot \color{blue}{i} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-251}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-227}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 7.8e-59)
   (fma (/ z a) a a)
   (if (<= y 7.5e+30) (+ (* (/ z y) y) (+ a t)) (+ (* i y) (+ a t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 7.8e-59) {
		tmp = fma((z / a), a, a);
	} else if (y <= 7.5e+30) {
		tmp = ((z / y) * y) + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 7.8e-59)
		tmp = fma(Float64(z / a), a, a);
	elseif (y <= 7.5e+30)
		tmp = Float64(Float64(Float64(z / y) * y) + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 7.8e-59], N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision], If[LessEqual[y, 7.5e+30], N[(N[(N[(z / y), $MachinePrecision] * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.80000000000000038e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 30.4%

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

   if 7.80000000000000038e-59 < y < 7.49999999999999973e30

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 55.3%

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

   if 7.49999999999999973e30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 68.7%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{y} \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -2e+136) (fma (/ z a) a a) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2e+136) {
		tmp = fma((z / a), a, a);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2e+136)
		tmp = fma(Float64(z / a), a, a);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2e+136], N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000012e136

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.1%

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

   if -2.00000000000000012e136 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 85.7

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

5. Applied rewrites 85.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 53.7%

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

9. Final simplification 53.7%

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

Alternative 14: 24.2% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.8

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

5. Applied rewrites 27.8%

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

6. Final simplification 27.8%

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

Reproduce

herbie shell --seed 2024255 
(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)))