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

Percentage Accurate: 99.8% → 99.8%

Time: 13.4s

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: 52.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + t\_1\right)\right)\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+39}:\\ \;\;\;\;\log c \cdot b + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z t_1)))))))
   (if (<= t_2 -2e+39)
     (+ (* (log c) b) (fma y i z))
     (if (<= t_2 5e+242) (+ t_1 (+ 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 t_1 = log(y) * x;
	double t_2 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	double tmp;
	if (t_2 <= -2e+39) {
		tmp = (log(c) * b) + fma(y, i, z);
	} else if (t_2 <= 5e+242) {
		tmp = t_1 + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	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)))))
	tmp = 0.0
	if (t_2 <= -2e+39)
		tmp = Float64(Float64(log(c) * b) + fma(y, i, z));
	elseif (t_2 <= 5e+242)
		tmp = Float64(t_1 + 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_] := 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]}, If[LessEqual[t$95$2, -2e+39], N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+242], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.99999999999999988e39

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 50.3%

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

   if -1.99999999999999988e39 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5.0000000000000004e242

   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 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.8%

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

   if 5.0000000000000004e242 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. 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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.0%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -2 \cdot 10^{+39}:\\ \;\;\;\;\log c \cdot b + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\log y \cdot x + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + t\_1\right)\right)\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z t_1)))))))
   (if (<= t_2 -2e+39)
     (* (+ (/ z i) y) i)
     (if (<= t_2 5e+242) (+ t_1 (+ 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 t_1 = log(y) * x;
	double t_2 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	double tmp;
	if (t_2 <= -2e+39) {
		tmp = ((z / i) + y) * i;
	} else if (t_2 <= 5e+242) {
		tmp = t_1 + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = (i * y) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + t_1))))
    if (t_2 <= (-2d+39)) then
        tmp = ((z / i) + y) * i
    else if (t_2 <= 5d+242) then
        tmp = t_1 + (a + t)
    else
        tmp = (i * y) + (a + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = Math.log(y) * x;
	double t_2 = (i * y) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	double tmp;
	if (t_2 <= -2e+39) {
		tmp = ((z / i) + y) * i;
	} else if (t_2 <= 5e+242) {
		tmp = t_1 + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = math.log(y) * x
	t_2 = (i * y) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + t_1))))
	tmp = 0
	if t_2 <= -2e+39:
		tmp = ((z / i) + y) * i
	elif t_2 <= 5e+242:
		tmp = t_1 + (a + t)
	else:
		tmp = (i * y) + (a + t)
	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)))))
	tmp = 0.0
	if (t_2 <= -2e+39)
		tmp = Float64(Float64(Float64(z / i) + y) * i);
	elseif (t_2 <= 5e+242)
		tmp = Float64(t_1 + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = log(y) * x;
	t_2 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + t_1))));
	tmp = 0.0;
	if (t_2 <= -2e+39)
		tmp = ((z / i) + y) * i;
	elseif (t_2 <= 5e+242)
		tmp = t_1 + (a + t);
	else
		tmp = (i * y) + (a + t);
	end
	tmp_2 = 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]}, If[LessEqual[t$95$2, -2e+39], N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$2, 5e+242], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.99999999999999988e39

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 26.9%

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

   if -1.99999999999999988e39 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5.0000000000000004e242

   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 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.8%

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

   if 5.0000000000000004e242 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. 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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.0%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -2 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\log y \cdot x + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* i y)
          (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))))
   (if (<= t_1 -2e+63)
     (* (+ (/ z i) y) i)
     (if (<= t_1 5e+246) (+ (* (log c) b) (+ 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 t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double tmp;
	if (t_1 <= -2e+63) {
		tmp = ((z / i) + y) * i;
	} else if (t_1 <= 5e+246) {
		tmp = (log(c) * b) + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * y) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))
    if (t_1 <= (-2d+63)) then
        tmp = ((z / i) + y) * i
    else if (t_1 <= 5d+246) then
        tmp = (log(c) * b) + (a + t)
    else
        tmp = (i * y) + (a + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))));
	double tmp;
	if (t_1 <= -2e+63) {
		tmp = ((z / i) + y) * i;
	} else if (t_1 <= 5e+246) {
		tmp = (Math.log(c) * b) + (a + t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (i * y) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))
	tmp = 0
	if t_1 <= -2e+63:
		tmp = ((z / i) + y) * i
	elif t_1 <= 5e+246:
		tmp = (math.log(c) * b) + (a + t)
	else:
		tmp = (i * y) + (a + t)
	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 <= -2e+63)
		tmp = Float64(Float64(Float64(z / i) + y) * i);
	elseif (t_1 <= 5e+246)
		tmp = Float64(Float64(log(c) * b) + Float64(a + t));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	tmp = 0.0;
	if (t_1 <= -2e+63)
		tmp = ((z / i) + y) * i;
	elseif (t_1 <= 5e+246)
		tmp = (log(c) * b) + (a + t);
	else
		tmp = (i * y) + (a + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[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, -2e+63], N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+246], N[(N[(N[Log[c], $MachinePrecision] * b), $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 (+.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)) < -2.00000000000000012e63

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 27.3%

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

   if -2.00000000000000012e63 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.99999999999999976e246

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 49.4%

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

   if 4.99999999999999976e246 < (+.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 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -9 \cdot 10^{+303}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* i y)
          (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))))
   (if (<= t_1 -9e+303)
     (* i y)
     (if (<= t_1 -200.0) (* (/ z i) i) (+ (* i y) (+ a t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double tmp;
	if (t_1 <= -9e+303) {
		tmp = i * y;
	} else if (t_1 <= -200.0) {
		tmp = (z / i) * i;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * y) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))
    if (t_1 <= (-9d+303)) then
        tmp = i * y
    else if (t_1 <= (-200.0d0)) then
        tmp = (z / i) * i
    else
        tmp = (i * y) + (a + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (i * y) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))));
	double tmp;
	if (t_1 <= -9e+303) {
		tmp = i * y;
	} else if (t_1 <= -200.0) {
		tmp = (z / i) * i;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (i * y) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))
	tmp = 0
	if t_1 <= -9e+303:
		tmp = i * y
	elif t_1 <= -200.0:
		tmp = (z / i) * i
	else:
		tmp = (i * y) + (a + t)
	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 <= -9e+303)
		tmp = Float64(i * y);
	elseif (t_1 <= -200.0)
		tmp = Float64(Float64(z / i) * i);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	tmp = 0.0;
	if (t_1 <= -9e+303)
		tmp = i * y;
	elseif (t_1 <= -200.0)
		tmp = (z / i) * i;
	else
		tmp = (i * y) + (a + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y), $MachinePrecision] + N[(N[(N[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, -9e+303], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -200.0], N[(N[(z / i), $MachinePrecision] * i), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -8.9999999999999997e303

   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 75.1

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

   5. Applied rewrites 75.1%

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

   if -8.9999999999999997e303 < (+.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)) < -200

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 9.3%

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

   if -200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 79.7

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

   5. Applied rewrites 79.7%

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

   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 49.9%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -9 \cdot 10^{+303}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -200:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 10^{+54}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))
      1e+54)
   (* (+ (/ z i) y) i)
   (+ (* 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 (((i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= 1e+54) {
		tmp = ((z / i) + y) * i;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((i * y) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))) <= 1d+54) then
        tmp = ((z / i) + y) * i
    else
        tmp = (i * y) + (a + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((i * y) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))))) <= 1e+54) {
		tmp = ((z / i) + y) * i;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((i * y) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))) <= 1e+54:
		tmp = ((z / i) + y) * i
	else:
		tmp = (i * y) + (a + t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(i * y) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))))) <= 1e+54)
		tmp = Float64(Float64(Float64(z / i) + y) * i);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((i * y) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= 1e+54)
		tmp = ((z / i) + y) * i;
	else
		tmp = (i * y) + (a + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(i * y), $MachinePrecision] + N[(N[(N[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], 1e+54], N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.0000000000000001e54

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 26.2%

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

   if 1.0000000000000001e54 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   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 50.3%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq 10^{+54}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ a t))))
   (if (<= x -3.1e+113)
     t_1
     (if (<= x 7e+119)
       (+ (+ (+ (fma (- b 0.5) (log c) z) t) a) (* i y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma((b - 0.5), log(c), fma(log(y), x, z)) + (a + t);
	double tmp;
	if (x <= -3.1e+113) {
		tmp = t_1;
	} else if (x <= 7e+119) {
		tmp = ((fma((b - 0.5), log(c), z) + t) + a) + (i * y);
	} 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 <= -3.1e+113)
		tmp = t_1;
	elseif (x <= 7e+119)
		tmp = Float64(Float64(Float64(fma(Float64(b - 0.5), log(c), z) + t) + a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(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, -3.1e+113], t$95$1, If[LessEqual[x, 7e+119], N[(N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.09999999999999991e113 or 7.0000000000000001e119 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 -3.09999999999999991e113 < x < 7.0000000000000001e119

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.1e+116)
   (+ (fma (- b 0.5) (log c) (fma (log y) x t)) (fma y i z))
   (+ (+ (+ (fma (- b 0.5) (log c) z) t) a) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.1e+116) {
		tmp = fma((b - 0.5), log(c), fma(log(y), x, t)) + fma(y, i, z);
	} else {
		tmp = ((fma((b - 0.5), log(c), z) + t) + a) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.1e+116)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, t)) + fma(y, i, z));
	else
		tmp = Float64(Float64(Float64(fma(Float64(b - 0.5), log(c), z) + t) + a) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.1e+116], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + t), $MachinePrecision]), $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1000000000000001e116

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if 2.1000000000000001e116 < a

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(i * y), $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 t around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 83.0

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

5. Applied rewrites 83.0%

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

6. Final simplification 83.0%

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

Alternative 10: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log y) x) (fma y i z))))
   (if (<= x -1.15e+144)
     t_1
     (if (<= x 1.8e+119)
       (+ (+ (+ (fma (- b 0.5) (log c) z) t) a) (* i y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (log(y) * x) + fma(y, i, z);
	double tmp;
	if (x <= -1.15e+144) {
		tmp = t_1;
	} else if (x <= 1.8e+119) {
		tmp = ((fma((b - 0.5), log(c), z) + t) + a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(log(y) * x) + fma(y, i, z))
	tmp = 0.0
	if (x <= -1.15e+144)
		tmp = t_1;
	elseif (x <= 1.8e+119)
		tmp = Float64(Float64(Float64(fma(Float64(b - 0.5), log(c), z) + t) + a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+144], t$95$1, If[LessEqual[x, 1.8e+119], N[(N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e144 or 1.80000000000000001e119 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.5%

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

   if -1.1500000000000001e144 < x < 1.80000000000000001e119

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log y) x) (fma y i z))))
   (if (<= x -1.15e+144)
     t_1
     (if (<= x 1.8e+119)
       (+ (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) + fma(y, i, z);
	double tmp;
	if (x <= -1.15e+144) {
		tmp = t_1;
	} else if (x <= 1.8e+119) {
		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) + fma(y, i, z))
	tmp = 0.0
	if (x <= -1.15e+144)
		tmp = t_1;
	elseif (x <= 1.8e+119)
		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[(y * i + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+144], t$95$1, If[LessEqual[x, 1.8e+119], 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 < -1.1500000000000001e144 or 1.80000000000000001e119 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.5%

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

   if -1.1500000000000001e144 < x < 1.80000000000000001e119

   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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right) + \mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log y) x) (fma y i z))))
   (if (<= x -1.15e+144)
     t_1
     (if (<= x 1.4e+118) (+ (fma y i z) (fma (- b 0.5) (log c) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (log(y) * x) + fma(y, i, z);
	double tmp;
	if (x <= -1.15e+144) {
		tmp = t_1;
	} else if (x <= 1.4e+118) {
		tmp = fma(y, i, z) + fma((b - 0.5), log(c), a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(log(y) * x) + fma(y, i, z))
	tmp = 0.0
	if (x <= -1.15e+144)
		tmp = t_1;
	elseif (x <= 1.4e+118)
		tmp = Float64(fma(y, i, z) + fma(Float64(b - 0.5), log(c), a));
	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[(y * i + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+144], t$95$1, If[LessEqual[x, 1.4e+118], N[(N[(y * i + z), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e144 or 1.39999999999999993e118 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.5%

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

   if -1.1500000000000001e144 < x < 1.39999999999999993e118

   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right) + \mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x + \mathsf{fma}\left(y, i, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.4% 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 81.0

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

5. Applied rewrites 81.0%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 50.9%

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

9. Final simplification 50.9%

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

Alternative 14: 23.5% 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 20.8

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

5. Applied rewrites 20.8%

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

6. Final simplification 20.8%

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

Reproduce

herbie shell --seed 2024243 
(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)))