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

Percentage Accurate: 99.8% → 99.8%

Time: 15.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate-+l+ N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+173}:\\ \;\;\;\;a + t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+175}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (log c) (- b 0.5))))
   (if (<= t_2 -5e+173)
     (+ a t_1)
     (if (<= t_2 1e+175) (+ (+ z t) (+ a (* y i))) (fma y i t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = log(c) * (b - 0.5);
	double tmp;
	if (t_2 <= -5e+173) {
		tmp = a + t_1;
	} else if (t_2 <= 1e+175) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = fma(y, i, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(log(c) * Float64(b - 0.5))
	tmp = 0.0
	if (t_2 <= -5e+173)
		tmp = Float64(a + t_1);
	elseif (t_2 <= 1e+175)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(y * i)));
	else
		tmp = fma(y, i, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+173], N[(a + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+175], N[(N[(z + t), $MachinePrecision] + N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -5.00000000000000034e173

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 62.0%

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

   if -5.00000000000000034e173 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999994e174

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 73.8%

      \[\leadsto \left(t + z\right) + \left(y \cdot i + a\right) \]

   if 9.9999999999999994e174 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 70.0

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

   7. Applied rewrites 70.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log c \cdot \left(b - 0.5\right) \leq -5 \cdot 10^{+173}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{elif}\;\log c \cdot \left(b - 0.5\right) \leq 10^{+175}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right)\right) \leq -200:\\ \;\;\;\;\left(z + t\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* x (log y)))))))
      -200.0)
   (+ (+ z t) (* y i))
   (+ a (* y i))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((y * i) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (x * math.log(y))))))) <= -200.0:
		tmp = (z + t) + (y * i)
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(x * log(y))))))) <= -200.0)
		tmp = Float64(Float64(z + t) + Float64(y * i));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -200.0], N[(N[(z + t), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 43.7%

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

   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 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 38.8%

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

4. Final simplification 41.3%

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

Alternative 4: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 84.0

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

5. Applied rewrites 84.0%

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

Alternative 5: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 10^{+144}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ t (fma (log c) (+ b -0.5) z)))))
   (if (<= (- b 0.5) -1e+210)
     t_1
     (if (<= (- b 0.5) 1e+144)
       (+ (+ z t) (+ a (fma i y (* x (log 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 = a + (t + fma(log(c), (b + -0.5), z));
	double tmp;
	if ((b - 0.5) <= -1e+210) {
		tmp = t_1;
	} else if ((b - 0.5) <= 1e+144) {
		tmp = (z + t) + (a + fma(i, y, (x * log(y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(t + fma(log(c), Float64(b + -0.5), z)))
	tmp = 0.0
	if (Float64(b - 0.5) <= -1e+210)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 1e+144)
		tmp = Float64(Float64(z + t) + Float64(a + fma(i, y, Float64(x * log(y)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(t + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -1e+210], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 1e+144], N[(N[(z + t), $MachinePrecision] + N[(a + N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -9.99999999999999927e209 or 1.00000000000000002e144 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.7%

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

   if -9.99999999999999927e209 < (-.f64 b #s(literal 1/2 binary64)) < 1.00000000000000002e144

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 94.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{elif}\;b - 0.5 \leq 10^{+144}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;\left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ z t) (+ a (fma i y (* x (log y)))))))
   (if (<= x -1.4e+95)
     t_1
     (if (<= x 3.6e+109)
       (+ (+ t (+ a (fma (log c) (+ b -0.5) z))) (* y i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z + t) + (a + fma(i, y, (x * log(y))));
	double tmp;
	if (x <= -1.4e+95) {
		tmp = t_1;
	} else if (x <= 3.6e+109) {
		tmp = (t + (a + fma(log(c), (b + -0.5), z))) + (y * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z + t) + Float64(a + fma(i, y, Float64(x * log(y)))))
	tmp = 0.0
	if (x <= -1.4e+95)
		tmp = t_1;
	elseif (x <= 3.6e+109)
		tmp = Float64(Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z))) + Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z + t), $MachinePrecision] + N[(a + N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+95], t$95$1, If[LessEqual[x, 3.6e+109], N[(N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e95 or 3.6e109 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 92.1%

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

   if -1.3999999999999999e95 < x < 3.6e109

   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. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;\left(t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ z t) (+ a (fma i y (* x (log y)))))))
   (if (<= x -1.4e+95)
     t_1
     (if (<= x 3.6e+109)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z + t) + (a + fma(i, y, (x * log(y))));
	double tmp;
	if (x <= -1.4e+95) {
		tmp = t_1;
	} else if (x <= 3.6e+109) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z + t) + Float64(a + fma(i, y, Float64(x * log(y)))))
	tmp = 0.0
	if (x <= -1.4e+95)
		tmp = t_1;
	elseif (x <= 3.6e+109)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z + t), $MachinePrecision] + N[(a + N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+95], t$95$1, If[LessEqual[x, 3.6e+109], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e95 or 3.6e109 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 92.1%

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

   if -1.3999999999999999e95 < x < 3.6e109

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + t\right) + \left(a + \mathsf{fma}\left(i, y, x \cdot \log y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + t\right) + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, a\right)\right)\\ \mathbf{if}\;i \leq -1.4 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\left(z + t\right) + \left(a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ z t) (fma y i (fma (log c) -0.5 a)))))
   (if (<= i -1.4e+70)
     t_1
     (if (<= i 3.6e+54) (+ (+ z t) (+ a (* x (log 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 = (z + t) + fma(y, i, fma(log(c), -0.5, a));
	double tmp;
	if (i <= -1.4e+70) {
		tmp = t_1;
	} else if (i <= 3.6e+54) {
		tmp = (z + t) + (a + (x * log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z + t) + fma(y, i, fma(log(c), -0.5, a)))
	tmp = 0.0
	if (i <= -1.4e+70)
		tmp = t_1;
	elseif (i <= 3.6e+54)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(x * log(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[(z + t), $MachinePrecision] + N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.4e+70], t$95$1, If[LessEqual[i, 3.6e+54], N[(N[(z + t), $MachinePrecision] + N[(a + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.39999999999999995e70 or 3.6000000000000001e54 < 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 b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.0%

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

   if -1.39999999999999995e70 < i < 3.6000000000000001e54

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.3%

      \[\leadsto \left(t + z\right) + \left(x \cdot \log y + a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{+70}:\\ \;\;\;\;\left(z + t\right) + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, a\right)\right)\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\left(z + t\right) + \left(a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + t\right) + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{if}\;i \leq -1.4 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\left(z + t\right) + \left(a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ z t) (+ a (* y i)))))
   (if (<= i -1.4e+70)
     t_1
     (if (<= i 3.6e+54) (+ (+ z t) (+ a (* x (log 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 = (z + t) + (a + (y * i));
	double tmp;
	if (i <= -1.4e+70) {
		tmp = t_1;
	} else if (i <= 3.6e+54) {
		tmp = (z + t) + (a + (x * log(y)));
	} else {
		tmp = t_1;
	}
	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 = (z + t) + (a + (y * i))
    if (i <= (-1.4d+70)) then
        tmp = t_1
    else if (i <= 3.6d+54) then
        tmp = (z + t) + (a + (x * log(y)))
    else
        tmp = t_1
    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 = (z + t) + (a + (y * i));
	double tmp;
	if (i <= -1.4e+70) {
		tmp = t_1;
	} else if (i <= 3.6e+54) {
		tmp = (z + t) + (a + (x * Math.log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z + t) + (a + (y * i))
	tmp = 0
	if i <= -1.4e+70:
		tmp = t_1
	elif i <= 3.6e+54:
		tmp = (z + t) + (a + (x * math.log(y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z + t) + Float64(a + Float64(y * i)))
	tmp = 0.0
	if (i <= -1.4e+70)
		tmp = t_1;
	elseif (i <= 3.6e+54)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(x * log(y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z + t) + (a + (y * i));
	tmp = 0.0;
	if (i <= -1.4e+70)
		tmp = t_1;
	elseif (i <= 3.6e+54)
		tmp = (z + t) + (a + (x * log(y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z + t), $MachinePrecision] + N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.4e+70], t$95$1, If[LessEqual[i, 3.6e+54], N[(N[(z + t), $MachinePrecision] + N[(a + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.39999999999999995e70 or 3.6000000000000001e54 < 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 b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 80.0%

      \[\leadsto \left(t + z\right) + \left(y \cdot i + a\right) \]

   if -1.39999999999999995e70 < i < 3.6000000000000001e54

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.3%

      \[\leadsto \left(t + z\right) + \left(x \cdot \log y + a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{+70}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\left(z + t\right) + \left(a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + b \cdot \log c\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 10^{+164}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (* b (log c)))))
   (if (<= (- b 0.5) -1e+210)
     t_1
     (if (<= (- b 0.5) 1e+164) (+ (+ z t) (+ a (* y i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (b * log(c));
	double tmp;
	if ((b - 0.5) <= -1e+210) {
		tmp = t_1;
	} else if ((b - 0.5) <= 1e+164) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	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 = a + (b * log(c))
    if ((b - 0.5d0) <= (-1d+210)) then
        tmp = t_1
    else if ((b - 0.5d0) <= 1d+164) then
        tmp = (z + t) + (a + (y * i))
    else
        tmp = t_1
    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 = a + (b * Math.log(c));
	double tmp;
	if ((b - 0.5) <= -1e+210) {
		tmp = t_1;
	} else if ((b - 0.5) <= 1e+164) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = a + (b * math.log(c))
	tmp = 0
	if (b - 0.5) <= -1e+210:
		tmp = t_1
	elif (b - 0.5) <= 1e+164:
		tmp = (z + t) + (a + (y * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(b * log(c)))
	tmp = 0.0
	if (Float64(b - 0.5) <= -1e+210)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 1e+164)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = a + (b * log(c));
	tmp = 0.0;
	if ((b - 0.5) <= -1e+210)
		tmp = t_1;
	elseif ((b - 0.5) <= 1e+164)
		tmp = (z + t) + (a + (y * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -1e+210], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 1e+164], N[(N[(z + t), $MachinePrecision] + N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -9.99999999999999927e209 or 1e164 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 68.6%

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

   if -9.99999999999999927e209 < (-.f64 b #s(literal 1/2 binary64)) < 1e164

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 73.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{elif}\;b - 0.5 \leq 10^{+164}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;a + b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))))
   (if (<= (- b 0.5) -1e+230)
     t_1
     (if (<= (- b 0.5) 2e+173) (+ (+ z t) (+ a (* y i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double tmp;
	if ((b - 0.5) <= -1e+230) {
		tmp = t_1;
	} else if ((b - 0.5) <= 2e+173) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	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 = b * log(c)
    if ((b - 0.5d0) <= (-1d+230)) then
        tmp = t_1
    else if ((b - 0.5d0) <= 2d+173) then
        tmp = (z + t) + (a + (y * i))
    else
        tmp = t_1
    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 = b * Math.log(c);
	double tmp;
	if ((b - 0.5) <= -1e+230) {
		tmp = t_1;
	} else if ((b - 0.5) <= 2e+173) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = b * math.log(c)
	tmp = 0
	if (b - 0.5) <= -1e+230:
		tmp = t_1
	elif (b - 0.5) <= 2e+173:
		tmp = (z + t) + (a + (y * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	tmp = 0.0
	if (Float64(b - 0.5) <= -1e+230)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 2e+173)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = b * log(c);
	tmp = 0.0;
	if ((b - 0.5) <= -1e+230)
		tmp = t_1;
	elseif ((b - 0.5) <= 2e+173)
		tmp = (z + t) + (a + (y * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -1e+230], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 2e+173], N[(N[(z + t), $MachinePrecision] + N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -1.0000000000000001e230 or 2e173 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -1.0000000000000001e230 < (-.f64 b #s(literal 1/2 binary64)) < 2e173

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 72.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+230}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;b - 0.5 \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.4e+246)
     t_1
     (if (<= x 1.7e+240) (+ (+ z t) (+ a (* y i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.4e+246) {
		tmp = t_1;
	} else if (x <= 1.7e+240) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-3.4d+246)) then
        tmp = t_1
    else if (x <= 1.7d+240) then
        tmp = (z + t) + (a + (y * i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -3.4e+246) {
		tmp = t_1;
	} else if (x <= 1.7e+240) {
		tmp = (z + t) + (a + (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -3.4e+246:
		tmp = t_1
	elif x <= 1.7e+240:
		tmp = (z + t) + (a + (y * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.4e+246)
		tmp = t_1;
	elseif (x <= 1.7e+240)
		tmp = Float64(Float64(z + t) + Float64(a + Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3.4e+246)
		tmp = t_1;
	elseif (x <= 1.7e+240)
		tmp = (z + t) + (a + (y * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e+246], t$95$1, If[LessEqual[x, 1.7e+240], N[(N[(z + t), $MachinePrecision] + N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999988e246 or 1.70000000000000004e240 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -3.39999999999999988e246 < x < 1.70000000000000004e240

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 68.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+246}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;\left(z + t\right) + \left(a + y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.7% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-log.f64 82.5

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

5. Applied rewrites 82.5%

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

6. Taylor expanded in i around inf

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

8. Applied rewrites 63.1%

   \[\leadsto \left(t + z\right) + \left(y \cdot i + a\right) \]

9. Final simplification 63.1%

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

Alternative 14: 39.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 81.1

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

5. Applied rewrites 81.1%

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

6. Taylor expanded in i around inf

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

8. Applied rewrites 38.4%

   \[\leadsto a + y \cdot \color{blue}{i} \]
9. Add Preprocessing

Alternative 15: 24.2% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 22.7

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

5. Applied rewrites 22.7%

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

6. Final simplification 22.7%

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

Reproduce

herbie shell --seed 2024220 
(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)))