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

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[(z + t), $MachinePrecision] + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $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. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.9

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

   16. lift--.f64 N/A

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

   17. sub-neg N/A

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

   18. lower-+.f64 N/A

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

   19. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 16.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -10:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))))
   (if (<= t_1 -1e+308)
     (* y i)
     (if (<= t_1 -10.0)
       (* i (/ z i))
       (if (<= t_1 INFINITY) (* 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 = (y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = y * i;
	} else if (t_1 <= -10.0) {
		tmp = i * (z / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = y * i;
	} else if (t_1 <= -10.0) {
		tmp = i * (z / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = i * (a / i);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * i) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))
	tmp = 0
	if t_1 <= -1e+308:
		tmp = y * i
	elif t_1 <= -10.0:
		tmp = i * (z / i)
	elif t_1 <= math.inf:
		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(y * i) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x))))))
	tmp = 0.0
	if (t_1 <= -1e+308)
		tmp = Float64(y * i);
	elseif (t_1 <= -10.0)
		tmp = Float64(i * Float64(z / i));
	elseif (t_1 <= Inf)
		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 = (y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	tmp = 0.0;
	if (t_1 <= -1e+308)
		tmp = y * i;
	elseif (t_1 <= -10.0)
		tmp = i * (z / i);
	elseif (t_1 <= Inf)
		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[(y * i), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+308], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -10.0], N[(i * N[(z / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], 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)) < -1e308 or +inf.0 < (+.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 79.1

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

   5. Applied rewrites 79.1%

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

   if -1e308 < (+.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)) < -10

   1. Initial program 99.8%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 13.4%

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

   if -10 < (+.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)) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 5.5%

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

4. Final simplification 13.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+308}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -10:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{elif}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq \infty:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 28.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+53}:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(y + \frac{a}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))))
   (if (<= t_1 -1e+308)
     (* y i)
     (if (<= t_1 -1e+53) (* i (/ 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 t_1 = (y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = y * i;
	} else if (t_1 <= -1e+53) {
		tmp = i * (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) :: t_1
    real(8) :: tmp
    t_1 = (y * i) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))
    if (t_1 <= (-1d+308)) then
        tmp = y * i
    else if (t_1 <= (-1d+53)) then
        tmp = i * (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 t_1 = (y * i) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = y * i;
	} else if (t_1 <= -1e+53) {
		tmp = i * (z / i);
	} else {
		tmp = i * (y + (a / i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * i) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))
	tmp = 0
	if t_1 <= -1e+308:
		tmp = y * i
	elif t_1 <= -1e+53:
		tmp = i * (z / i)
	else:
		tmp = i * (y + (a / i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x))))))
	tmp = 0.0
	if (t_1 <= -1e+308)
		tmp = Float64(y * i);
	elseif (t_1 <= -1e+53)
		tmp = Float64(i * 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)
	t_1 = (y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))));
	tmp = 0.0;
	if (t_1 <= -1e+308)
		tmp = y * i;
	elseif (t_1 <= -1e+53)
		tmp = i * (z / i);
	else
		tmp = i * (y + (a / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+308], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -1e+53], N[(i * N[(z / i), $MachinePrecision]), $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)) < -1e308

   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 79.1

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

   5. Applied rewrites 79.1%

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

   if -1e308 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -9.9999999999999999e52

   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 68.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 13.6%

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

   if -9.9999999999999999e52 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 66.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 27.8%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+308}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -1 \cdot 10^{+53}:\\ \;\;\;\;i \cdot \frac{z}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(y + \frac{a}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, b \cdot \log c\right)\\ t_2 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+144}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* b (log c)))) (t_2 (* (log c) (- b 0.5))))
   (if (<= t_2 -1e+157) t_1 (if (<= t_2 2e+144) (* i (+ y (/ z i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (b * log(c)));
	double t_2 = log(c) * (b - 0.5);
	double tmp;
	if (t_2 <= -1e+157) {
		tmp = t_1;
	} else if (t_2 <= 2e+144) {
		tmp = i * (y + (z / i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(b * log(c)))
	t_2 = Float64(log(c) * Float64(b - 0.5))
	tmp = 0.0
	if (t_2 <= -1e+157)
		tmp = t_1;
	elseif (t_2 <= 2e+144)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	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[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+157], t$95$1, If[LessEqual[t$95$2, 2e+144], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -9.99999999999999983e156 or 2.00000000000000005e144 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -9.99999999999999983e156 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 2.00000000000000005e144

   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 69.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log c \cdot \left(b - 0.5\right) \leq -1 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;\log c \cdot \left(b - 0.5\right) \leq 2 \cdot 10^{+144}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))
      -10.0)
   (* i (+ y (/ z i)))
   (+ (* y i) (* x (/ a x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= -10.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = (y * i) + (x * (a / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((y * i) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))) <= (-10.0d0)) then
        tmp = i * (y + (z / i))
    else
        tmp = (y * i) + (x * (a / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))))) <= -10.0) {
		tmp = i * (y + (z / i));
	} else {
		tmp = (y * i) + (x * (a / x));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((y * i) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))) <= -10.0:
		tmp = i * (y + (z / i))
	else:
		tmp = (y * i) + (x * (a / x))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))))) <= -10.0)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	else
		tmp = Float64(Float64(y * i) + Float64(x * Float64(a / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= -10.0)
		tmp = i * (y + (z / i));
	else
		tmp = (y * i) + (x * (a / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -10.0], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(x * N[(a / x), $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)) < -10

   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 72.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 31.3%

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

   if -10 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 30.8%

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

4. Final simplification 31.0%

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

Alternative 6: 33.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))
      -10.0)
   (* 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 (((y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= -10.0) {
		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 (((y * i) + ((log(c) * (b - 0.5d0)) + (a + (t + (z + (log(y) * x)))))) <= (-10.0d0)) 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 (((y * i) + ((Math.log(c) * (b - 0.5)) + (a + (t + (z + (Math.log(y) * x)))))) <= -10.0) {
		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 ((y * i) + ((math.log(c) * (b - 0.5)) + (a + (t + (z + (math.log(y) * x)))))) <= -10.0:
		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(y * i) + Float64(Float64(log(c) * Float64(b - 0.5)) + Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))))) <= -10.0)
		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 (((y * i) + ((log(c) * (b - 0.5)) + (a + (t + (z + (log(y) * x)))))) <= -10.0)
		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[(y * i), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] + N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -10.0], 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)) < -10

   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 72.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 31.3%

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

   if -10 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 65.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 28.0%

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

4. Final simplification 29.7%

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

Alternative 7: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+99}:\\ \;\;\;\;a + \left(t\_1 + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) (+ b -0.5) z)))
   (if (<= x -7.5e+99)
     (+ a (+ t_1 (fma x (log y) t)))
     (if (<= x 3e+107)
       (+ t (+ a (fma y i t_1)))
       (fma (log y) x (fma y i (+ (+ z t) (* a 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), (b + -0.5), z);
	double tmp;
	if (x <= -7.5e+99) {
		tmp = a + (t_1 + fma(x, log(y), t));
	} else if (x <= 3e+107) {
		tmp = t + (a + fma(y, i, t_1));
	} else {
		tmp = fma(log(y), x, fma(y, i, ((z + t) + (a * 1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), Float64(b + -0.5), z)
	tmp = 0.0
	if (x <= -7.5e+99)
		tmp = Float64(a + Float64(t_1 + fma(x, log(y), t)));
	elseif (x <= 3e+107)
		tmp = Float64(t + Float64(a + fma(y, i, t_1)));
	else
		tmp = fma(log(y), x, fma(y, i, Float64(Float64(z + t) + Float64(a * 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[x, -7.5e+99], N[(a + N[(t$95$1 + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+107], N[(t + N[(a + N[(y * i + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(y * i + N[(N[(z + t), $MachinePrecision] + N[(a * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999963e99

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Applied rewrites 86.5%

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

   if -7.49999999999999963e99 < x < 3.00000000000000023e107

   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 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 97.9

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

   8. Applied rewrites 97.9%

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

   if 3.00000000000000023e107 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 90.7

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.3%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-+l+ N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 99.3

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

   12. Applied rewrites 99.3%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+99}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 83.7

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

5. Applied rewrites 83.7%

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

Alternative 9: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + t\right) + a \cdot 1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, t\_1\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ z t) (* a 1.0))))
   (if (<= x -1.45e+112)
     (+ (* y i) (fma (log y) x t_1))
     (if (<= x 3e+107)
       (+ t (+ a (fma y i (fma (log c) (+ b -0.5) z))))
       (fma (log y) x (fma y i t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z + t) + (a * 1.0);
	double tmp;
	if (x <= -1.45e+112) {
		tmp = (y * i) + fma(log(y), x, t_1);
	} else if (x <= 3e+107) {
		tmp = t + (a + fma(y, i, fma(log(c), (b + -0.5), z)));
	} else {
		tmp = fma(log(y), x, fma(y, i, t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z + t) + Float64(a * 1.0))
	tmp = 0.0
	if (x <= -1.45e+112)
		tmp = Float64(Float64(y * i) + fma(log(y), x, t_1));
	elseif (x <= 3e+107)
		tmp = Float64(t + Float64(a + fma(y, i, fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = fma(log(y), x, fma(y, i, t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z + t), $MachinePrecision] + N[(a * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+112], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+107], N[(t + N[(a + N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(y * i + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e112

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 86.9

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 88.1%

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

   if -1.4500000000000001e112 < x < 3.00000000000000023e107

   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 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

   if 3.00000000000000023e107 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 90.7

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.3%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-+l+ N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 99.3

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

   12. Applied rewrites 99.3%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, \left(z + t\right) + a \cdot 1\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log y) x (fma y i (+ (+ z t) (* a 1.0))))))
   (if (<= x -1.45e+112)
     t_1
     (if (<= x 3e+107) (+ t (+ a (fma y i (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(log(y), x, fma(y, i, ((z + t) + (a * 1.0))));
	double tmp;
	if (x <= -1.45e+112) {
		tmp = t_1;
	} else if (x <= 3e+107) {
		tmp = t + (a + fma(y, i, 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(log(y), x, fma(y, i, Float64(Float64(z + t) + Float64(a * 1.0))))
	tmp = 0.0
	if (x <= -1.45e+112)
		tmp = t_1;
	elseif (x <= 3e+107)
		tmp = Float64(t + Float64(a + fma(y, i, fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[(y * i + N[(N[(z + t), $MachinePrecision] + N[(a * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+112], t$95$1, If[LessEqual[x, 3e+107], N[(t + N[(a + N[(y * i + 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 < -1.4500000000000001e112 or 3.00000000000000023e107 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 88.7

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 93.2%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-+l+ N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 93.2

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

   12. Applied rewrites 93.2%

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

   if -1.4500000000000001e112 < x < 3.00000000000000023e107

   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 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(y, i, \left(z + t\right) + a \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \log y \cdot x\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (* (log y) x))))
   (if (<= x -3.8e+173)
     t_1
     (if (<= x 1.7e+213)
       (+ t (+ a (fma y i (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 = (y * i) + (log(y) * x);
	double tmp;
	if (x <= -3.8e+173) {
		tmp = t_1;
	} else if (x <= 1.7e+213) {
		tmp = t + (a + fma(y, i, fma(log(c), (b + -0.5), z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -3.8e+173)
		tmp = t_1;
	elseif (x <= 1.7e+213)
		tmp = Float64(t + Float64(a + fma(y, i, fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+173], t$95$1, If[LessEqual[x, 1.7e+213], N[(t + N[(a + N[(y * i + 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 < -3.80000000000000011e173 or 1.69999999999999996e213 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -3.80000000000000011e173 < x < 1.69999999999999996e213

   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 24.2

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 96.1

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

   8. Applied rewrites 96.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+173}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \log y \cdot x\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (* (log y) x))))
   (if (<= x -3.8e+173)
     t_1
     (if (<= x 1.7e+213)
       (+ 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 = (y * i) + (log(y) * x);
	double tmp;
	if (x <= -3.8e+173) {
		tmp = t_1;
	} else if (x <= 1.7e+213) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -3.8e+173)
		tmp = t_1;
	elseif (x <= 1.7e+213)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+173], t$95$1, If[LessEqual[x, 1.7e+213], 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 < -3.80000000000000011e173 or 1.69999999999999996e213 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -3.80000000000000011e173 < x < 1.69999999999999996e213

   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 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+173}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-224}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+141}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* b (log c)))))
   (if (<= b -3.1e+141)
     t_1
     (if (<= b 2.2e-224)
       (* i (+ y (/ z i)))
       (if (<= b 2.45e+141) (+ (* y i) (* (log y) x)) 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, (b * log(c)));
	double tmp;
	if (b <= -3.1e+141) {
		tmp = t_1;
	} else if (b <= 2.2e-224) {
		tmp = i * (y + (z / i));
	} else if (b <= 2.45e+141) {
		tmp = (y * i) + (log(y) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(b * log(c)))
	tmp = 0.0
	if (b <= -3.1e+141)
		tmp = t_1;
	elseif (b <= 2.2e-224)
		tmp = Float64(i * Float64(y + Float64(z / i)));
	elseif (b <= 2.45e+141)
		tmp = Float64(Float64(y * i) + Float64(log(y) * x));
	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[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.1e+141], t$95$1, If[LessEqual[b, 2.2e-224], N[(i * N[(y + N[(z / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+141], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.10000000000000004e141 or 2.45000000000000005e141 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.1

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

   5. Applied rewrites 71.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 71.1

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

   7. Applied rewrites 71.1%

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

   if -3.10000000000000004e141 < b < 2.2000000000000001e-224

   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 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 74.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 36.5%

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

   if 2.2000000000000001e-224 < b < 2.45000000000000005e141

   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

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 44.0

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

   5. Applied rewrites 44.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-224}:\\ \;\;\;\;i \cdot \left(y + \frac{z}{i}\right)\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+141}:\\ \;\;\;\;y \cdot i + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-79}:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 4.4e-79) (* 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 <= 4.4e-79) {
		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 <= 4.4d-79) 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 <= 4.4e-79) {
		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 <= 4.4e-79:
		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 <= 4.4e-79)
		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 <= 4.4e-79)
		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, 4.4e-79], N[(i * N[(a / i), $MachinePrecision]), $MachinePrecision], N[(y * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999998e-79

   1. Initial program 99.8%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 10.7%

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

   if 4.3999999999999998e-79 < 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 34.0

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

   5. Applied rewrites 34.0%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-79}:\\ \;\;\;\;i \cdot \frac{a}{i}\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 24.3% 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 22.7

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

5. Applied rewrites 22.7%

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

6. Final simplification 22.7%

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

Reproduce

herbie shell --seed 2024219 
(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)))