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

Percentage Accurate: 99.8% → 99.8%

Time: 14.7s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

   5. lift-+.f64 N/A

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

   6. 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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 36.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+302}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{t}, t\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ a (+ t (+ z (* x (log y))))) (* (log c) (- b 0.5)))
          (* y i))))
   (if (<= t_1 -1e+302)
     (* y i)
     (if (<= t_1 -100.0)
       (fma t (/ z t) t)
       (if (<= t_1 1.5e+306) (fma t (/ a t) t) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i);
	double tmp;
	if (t_1 <= -1e+302) {
		tmp = y * i;
	} else if (t_1 <= -100.0) {
		tmp = fma(t, (z / t), t);
	} else if (t_1 <= 1.5e+306) {
		tmp = fma(t, (a / t), t);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(t + Float64(z + Float64(x * log(y))))) + Float64(log(c) * Float64(b - 0.5))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -1e+302)
		tmp = Float64(y * i);
	elseif (t_1 <= -100.0)
		tmp = fma(t, Float64(z / t), t);
	elseif (t_1 <= 1.5e+306)
		tmp = fma(t, Float64(a / t), t);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+302], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(t * N[(z / t), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+306], N[(t * N[(a / t), $MachinePrecision] + t), $MachinePrecision], N[(y * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.0000000000000001e302 or 1.5000000000000001e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.0000000000000001e302 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 32.7%

      \[\leadsto \mathsf{fma}\left(t, \frac{z}{\color{blue}{t}}, t\right) \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.5000000000000001e306

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 23.2%

      \[\leadsto \mathsf{fma}\left(t, \frac{a}{\color{blue}{t}}, t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -1 \cdot 10^{+302}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -100:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{t}, t\right)\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq 1.5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 29.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ a (+ t (+ z (* x (log y))))) (* (log c) (- b 0.5)))
          (* y i))))
   (if (<= t_1 -2e+294)
     (* y i)
     (if (<= t_1 -100.0)
       (* i (/ z i))
       (if (<= t_1 1.5e+306) (fma t (/ a t) t) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i);
	double tmp;
	if (t_1 <= -2e+294) {
		tmp = y * i;
	} else if (t_1 <= -100.0) {
		tmp = i * (z / i);
	} else if (t_1 <= 1.5e+306) {
		tmp = fma(t, (a / t), t);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(t + Float64(z + Float64(x * log(y))))) + Float64(log(c) * Float64(b - 0.5))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+294)
		tmp = Float64(y * i);
	elseif (t_1 <= -100.0)
		tmp = Float64(i * Float64(z / i));
	elseif (t_1 <= 1.5e+306)
		tmp = fma(t, Float64(a / t), t);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+294], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(i * N[(z / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+306], N[(t * N[(a / t), $MachinePrecision] + t), $MachinePrecision], N[(y * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000013e294 or 1.5000000000000001e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -2.00000000000000013e294 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 13.1%

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

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.5000000000000001e306

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 23.2%

      \[\leadsto \mathsf{fma}\left(t, \frac{a}{\color{blue}{t}}, t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -100:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq 1.5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 22.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ a (+ t (+ z (* x (log y))))) (* (log c) (- b 0.5)))
          (* y i))))
   (if (<= t_1 -2e+294)
     (* y i)
     (if (<= t_1 -100.0)
       (* i (/ z i))
       (if (<= t_1 5e+282) (* i (/ a i)) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i);
	double tmp;
	if (t_1 <= -2e+294) {
		tmp = y * i;
	} else if (t_1 <= -100.0) {
		tmp = i * (z / i);
	} else if (t_1 <= 5e+282) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5d0))) + (y * i)
    if (t_1 <= (-2d+294)) then
        tmp = y * i
    else if (t_1 <= (-100.0d0)) then
        tmp = i * (z / i)
    else if (t_1 <= 5d+282) then
        tmp = i * (a / i)
    else
        tmp = y * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (t + (z + (x * Math.log(y))))) + (Math.log(c) * (b - 0.5))) + (y * i);
	double tmp;
	if (t_1 <= -2e+294) {
		tmp = y * i;
	} else if (t_1 <= -100.0) {
		tmp = i * (z / i);
	} else if (t_1 <= 5e+282) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((a + (t + (z + (x * math.log(y))))) + (math.log(c) * (b - 0.5))) + (y * i)
	tmp = 0
	if t_1 <= -2e+294:
		tmp = y * i
	elif t_1 <= -100.0:
		tmp = i * (z / i)
	elif t_1 <= 5e+282:
		tmp = i * (a / i)
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(t + Float64(z + Float64(x * log(y))))) + Float64(log(c) * Float64(b - 0.5))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+294)
		tmp = Float64(y * i);
	elseif (t_1 <= -100.0)
		tmp = Float64(i * Float64(z / i));
	elseif (t_1 <= 5e+282)
		tmp = Float64(i * Float64(a / i));
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -2e+294)
		tmp = y * i;
	elseif (t_1 <= -100.0)
		tmp = i * (z / i);
	elseif (t_1 <= 5e+282)
		tmp = i * (a / i);
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+294], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(i * N[(z / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+282], N[(i * N[(a / i), $MachinePrecision]), $MachinePrecision], N[(y * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000013e294 or 4.99999999999999978e282 < (+.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 y around inf

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

      1. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   if -2.00000000000000013e294 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 13.1%

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

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.99999999999999978e282

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 12.0%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -100:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq 5 \cdot 10^{+282}:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+302}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(y + \frac{a}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ a (+ t (+ z (* x (log y))))) (* (log c) (- b 0.5)))
          (* y i))))
   (if (<= t_1 -1e+302)
     (* y i)
     (if (<= t_1 -2e+67) (fma t (/ z t) t) (* i (+ y (/ a i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i);
	double tmp;
	if (t_1 <= -1e+302) {
		tmp = y * i;
	} else if (t_1 <= -2e+67) {
		tmp = fma(t, (z / t), t);
	} else {
		tmp = i * (y + (a / i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(t + Float64(z + Float64(x * log(y))))) + Float64(log(c) * Float64(b - 0.5))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -1e+302)
		tmp = Float64(y * i);
	elseif (t_1 <= -2e+67)
		tmp = fma(t, Float64(z / t), t);
	else
		tmp = Float64(i * Float64(y + Float64(a / i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+302], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -2e+67], N[(t * N[(z / t), $MachinePrecision] + t), $MachinePrecision], N[(i * N[(y + N[(a / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1.0000000000000001e302

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if -1.0000000000000001e302 < (+.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)) < -1.99999999999999997e67

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 33.3%

      \[\leadsto \mathsf{fma}\left(t, \frac{z}{\color{blue}{t}}, t\right) \]

   if -1.99999999999999997e67 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 35.9%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -1 \cdot 10^{+302}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(a + \left(t + \left(z + x \cdot \log y\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) + y \cdot i \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{t}, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(y + \frac{a}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double tmp;
	if ((((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i)) <= -100.0) {
		tmp = fma(y, i, ((z + t) + t_1));
	} else {
		tmp = fma(y, i, (a + 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(Float64(Float64(a + Float64(t + Float64(z + Float64(x * log(y))))) + Float64(log(c) * Float64(b - 0.5))) + Float64(y * i)) <= -100.0)
		tmp = fma(y, i, Float64(Float64(z + t) + t_1));
	else
		tmp = fma(y, i, Float64(a + 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]}, If[LessEqual[N[(N[(N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -100.0], N[(y * i + N[(N[(z + t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 92.8

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

   7. Applied rewrites 92.8%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 78.2

         \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   10. Applied rewrites 78.2%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 68.1%

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

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

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 84.2

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 52.7%

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

4. Final simplification 60.6%

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

Alternative 7: 32.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5))) + (y * i)) <= -100.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = i * (y + (a / 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 ((((a + (t + (z + (x * log(y))))) + (log(c) * (b - 0.5d0))) + (y * i)) <= (-100.0d0)) then
        tmp = i * (y + (z / i))
    else
        tmp = i * (y + (a / 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 ((((a + (t + (z + (x * Math.log(y))))) + (Math.log(c) * (b - 0.5))) + (y * i)) <= -100.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = i * (y + (a / i));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 30.5%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 35.5%

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

4. Final simplification 33.0%

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

Alternative 8: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{if}\;b - 0.5 \leq -5 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (+ a (fma (log c) (+ b -0.5) z)))))
   (if (<= (- b 0.5) -5e+150)
     t_1
     (if (<= (- b 0.5) 2e+172)
       (fma y i (+ (fma x (log y) (+ z t)) (* (- a) -1.0)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (a + fma(log(c), (b + -0.5), z)));
	double tmp;
	if ((b - 0.5) <= -5e+150) {
		tmp = t_1;
	} else if ((b - 0.5) <= 2e+172) {
		tmp = fma(y, i, (fma(x, log(y), (z + t)) + (-a * -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(a + fma(log(c), Float64(b + -0.5), z)))
	tmp = 0.0
	if (Float64(b - 0.5) <= -5e+150)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 2e+172)
		tmp = fma(y, i, Float64(fma(x, log(y), Float64(z + t)) + Float64(Float64(-a) * -1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -5e+150], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 2e+172], N[(y * i + N[(N[(x * N[Log[y], $MachinePrecision] + N[(z + t), $MachinePrecision]), $MachinePrecision] + N[((-a) * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -5.00000000000000009e150 or 2.0000000000000002e172 < (-.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. 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 x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 92.5%

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

   if -5.00000000000000009e150 < (-.f64 b #s(literal 1/2 binary64)) < 2.0000000000000002e172

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 97.0

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

   7. Applied rewrites 97.0%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(\mathsf{neg}\left(a\right)\right) \cdot -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.5%

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -1.25e+207)
     t_1
     (if (<= x 1.25e+253)
       (fma y i (+ a (+ t (fma (log c) (+ b -0.5) z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (x * log(y)));
	double tmp;
	if (x <= -1.25e+207) {
		tmp = t_1;
	} else if (x <= 1.25e+253) {
		tmp = fma(y, i, (a + (t + fma(log(c), (b + -0.5), z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -1.25e+207)
		tmp = t_1;
	elseif (x <= 1.25e+253)
		tmp = fma(y, i, Float64(a + Float64(t + fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+207], t$95$1, If[LessEqual[x, 1.25e+253], N[(y * i + N[(a + N[(t + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e207 or 1.2499999999999999e253 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -1.25e207 < x < 1.2499999999999999e253

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 94.6

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

   7. Applied rewrites 94.6%

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

Alternative 10: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+253}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -1.25e+207)
     t_1
     (if (<= x 1.25e+253)
       (+ 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 = fma(y, i, (x * log(y)));
	double tmp;
	if (x <= -1.25e+207) {
		tmp = t_1;
	} else if (x <= 1.25e+253) {
		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 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -1.25e+207)
		tmp = t_1;
	elseif (x <= 1.25e+253)
		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[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+207], t$95$1, If[LessEqual[x, 1.25e+253], 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 < -1.25e207 or 1.2499999999999999e253 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -1.25e207 < x < 1.2499999999999999e253

   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 94.6

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

   5. Applied rewrites 94.6%

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

Alternative 11: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -1.25e+207)
     t_1
     (if (<= x 1.25e+253) (fma y i (+ a (fma (log c) (+ b -0.5) z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (x * log(y)));
	double tmp;
	if (x <= -1.25e+207) {
		tmp = t_1;
	} else if (x <= 1.25e+253) {
		tmp = fma(y, i, (a + fma(log(c), (b + -0.5), z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -1.25e+207)
		tmp = t_1;
	elseif (x <= 1.25e+253)
		tmp = fma(y, i, Float64(a + fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+207], t$95$1, If[LessEqual[x, 1.25e+253], N[(y * i + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e207 or 1.2499999999999999e253 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -1.25e207 < x < 1.2499999999999999e253

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 77.8%

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

Alternative 12: 73.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -1.3e+205)
     t_1
     (if (<= x 3.1e+248) (fma y i (+ (* (- a) -1.0) (+ z t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (x * log(y)));
	double tmp;
	if (x <= -1.3e+205) {
		tmp = t_1;
	} else if (x <= 3.1e+248) {
		tmp = fma(y, i, ((-a * -1.0) + (z + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -1.3e+205)
		tmp = t_1;
	elseif (x <= 3.1e+248)
		tmp = fma(y, i, Float64(Float64(Float64(-a) * -1.0) + Float64(z + t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+205], t$95$1, If[LessEqual[x, 3.1e+248], N[(y * i + N[(N[((-a) * -1.0), $MachinePrecision] + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2999999999999999e205 or 3.10000000000000005e248 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 85.9

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

   7. Applied rewrites 85.9%

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

   if -1.2999999999999999e205 < x < 3.10000000000000005e248

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 90.1

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

   7. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 85.1

         \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   10. Applied rewrites 85.1%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(\mathsf{neg}\left(a\right)\right) \cdot -1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 77.0%

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(-a\right) \cdot -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + b \cdot \log c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -6.4e+109)
   (fma y i (+ (* (- a) -1.0) (+ z t)))
   (fma y i (+ a (* b (log c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -6.4e+109) {
		tmp = fma(y, i, ((-a * -1.0) + (z + t)));
	} else {
		tmp = fma(y, i, (a + (b * log(c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -6.4e+109)
		tmp = fma(y, i, Float64(Float64(Float64(-a) * -1.0) + Float64(z + t)));
	else
		tmp = fma(y, i, Float64(a + Float64(b * log(c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -6.4e+109], N[(y * i + N[(N[((-a) * -1.0), $MachinePrecision] + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4000000000000002e109

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 94.7

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

   7. Applied rewrites 94.7%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 89.8

         \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   10. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(\mathsf{neg}\left(a\right)\right) \cdot -1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 83.4%

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(-a\right) \cdot -1\right) \]

   if -6.4000000000000002e109 < z

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 54.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.25e+207)
     t_1
     (if (<= x 2.2e+253) (fma y i (+ (* (- a) -1.0) (+ z t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.25e+207) {
		tmp = t_1;
	} else if (x <= 2.2e+253) {
		tmp = fma(y, i, ((-a * -1.0) + (z + t)));
	} 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 <= -1.25e+207)
		tmp = t_1;
	elseif (x <= 2.2e+253)
		tmp = fma(y, i, Float64(Float64(Float64(-a) * -1.0) + Float64(z + t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+207], t$95$1, If[LessEqual[x, 2.2e+253], N[(y * i + N[(N[((-a) * -1.0), $MachinePrecision] + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e207 or 2.20000000000000006e253 < x

   1. Initial program 99.9%

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

   3. 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 80.0

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

   5. Applied rewrites 80.0%

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

   if -1.25e207 < x < 2.20000000000000006e253

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 89.8

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

   7. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 84.9

         \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   10. Applied rewrites 84.9%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(\mathsf{neg}\left(a\right)\right) \cdot -1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.4%

       \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(-a\right) \cdot -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 540000:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 540000.0) (* i (/ a i)) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 540000.0) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= 540000.0d0) then
        tmp = i * (a / i)
    else
        tmp = y * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 540000.0) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 540000.0:
		tmp = i * (a / i)
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 540000.0)
		tmp = Float64(i * Float64(a / i));
	else
		tmp = 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 <= 540000.0)
		tmp = i * (a / i);
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 540000.0], N[(i * N[(a / i), $MachinePrecision]), $MachinePrecision], N[(y * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e5

   1. Initial program 99.9%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 13.3%

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

   if 5.4e5 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 540000:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (fma y i (+ (* (- a) -1.0) (+ z t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(y, i, ((-a * -1.0) + (z + t)));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(y, i, Float64(Float64(Float64(-a) * -1.0) + Float64(z + t)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(N[((-a) * -1.0), $MachinePrecision] + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

   5. lift-+.f64 N/A

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

   6. 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 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in a around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. sub-neg N/A

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

   6. associate-/l* N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f64 90.8

       \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{\color{blue}{b + -0.5}}{a}, -1\right)\right) \]

7. Applied rewrites 90.8%

   \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, z + t\right) + \color{blue}{\left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)}\right) \]

8. Taylor expanded in x around 0

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

   1. lower-+.f64 76.4

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

10. Applied rewrites 76.4%

    \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\left(t + z\right)} + \left(-a\right) \cdot \mathsf{fma}\left(-\log c, \frac{b + -0.5}{a}, -1\right)\right) \]

11. Taylor expanded in a around inf

    \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(\mathsf{neg}\left(a\right)\right) \cdot -1\right) \]
12. Step-by-step derivation

13. Applied rewrites 68.6%

    \[\leadsto \mathsf{fma}\left(y, i, \left(t + z\right) + \left(-a\right) \cdot -1\right) \]

14. Final simplification 68.6%

    \[\leadsto \mathsf{fma}\left(y, i, \left(-a\right) \cdot -1 + \left(z + t\right)\right) \]
15. Add Preprocessing

Alternative 17: 23.9% 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 99.9%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 24.3

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

5. Applied rewrites 24.3%

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

6. Final simplification 24.3%

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

Reproduce

herbie shell --seed 2024233 
(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)))