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

Percentage Accurate: 99.8% → 99.8%

Time: 15.0s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 36.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(\frac{y \cdot i}{z}, z, z\right) \]

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in y around inf

      \[\leadsto i \cdot y + a \]
   10. Step-by-step derivation

   11. Applied rewrites 35.8%

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

4. Final simplification 35.8%

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

Alternative 3: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{if}\;i \leq -2.8 \cdot 10^{+31}:\\ \;\;\;\;t\_1 + a\\ \mathbf{elif}\;i \leq 5.6 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (- b 0.5) (log c) (fma y i z))))
   (if (<= i -2.8e+31)
     (+ t_1 a)
     (if (<= i 5.6e-24)
       (+ (fma (- b 0.5) (log c) (fma (log y) x z)) (+ a t))
       (+ (+ a t) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma((b - 0.5), log(c), fma(y, i, z));
	double tmp;
	if (i <= -2.8e+31) {
		tmp = t_1 + a;
	} else if (i <= 5.6e-24) {
		tmp = fma((b - 0.5), log(c), fma(log(y), x, z)) + (a + t);
	} else {
		tmp = (a + t) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(b - 0.5), log(c), fma(y, i, z))
	tmp = 0.0
	if (i <= -2.8e+31)
		tmp = Float64(t_1 + a);
	elseif (i <= 5.6e-24)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(log(y), x, z)) + Float64(a + t));
	else
		tmp = Float64(Float64(a + t) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.8e+31], N[(t$95$1 + a), $MachinePrecision], If[LessEqual[i, 5.6e-24], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.80000000000000017e31

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.1%

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

   if -2.80000000000000017e31 < i < 5.6000000000000003e-24

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 5.6000000000000003e-24 < i

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{elif}\;i \leq 5.6 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+220}:\\ \;\;\;\;\left(\frac{a}{x} + \log y\right) \cdot x + i \cdot y\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+246}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -5.5e+220)
   (+ (* (+ (/ a x) (log y)) x) (* i y))
   (if (<= x 6.3e+246)
     (+ (+ a t) (fma (- b 0.5) (log c) (fma y i z)))
     (fma y i (* (log y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -5.5e+220) {
		tmp = (((a / x) + log(y)) * x) + (i * y);
	} else if (x <= 6.3e+246) {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(y, i, z));
	} else {
		tmp = fma(y, i, (log(y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -5.5e+220)
		tmp = Float64(Float64(Float64(Float64(a / x) + log(y)) * x) + Float64(i * y));
	elseif (x <= 6.3e+246)
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(y, i, z)));
	else
		tmp = fma(y, i, Float64(log(y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -5.5e+220], N[(N[(N[(N[(a / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.3e+246], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4999999999999999e220

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 93.7%

      \[\leadsto \left(\frac{a}{x} + \log y\right) \cdot x + y \cdot i \]

   if -5.4999999999999999e220 < x < 6.3e246

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 6.3e246 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+220}:\\ \;\;\;\;\left(\frac{a}{x} + \log y\right) \cdot x + i \cdot y\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+246}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+246}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (log y) x))))
   (if (<= x -6e+220)
     t_1
     (if (<= x 6.3e+246)
       (+ (+ a t) (fma (- b 0.5) (log c) (fma y i z)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (log(y) * x));
	double tmp;
	if (x <= -6e+220) {
		tmp = t_1;
	} else if (x <= 6.3e+246) {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(y, i, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (x <= -6e+220)
		tmp = t_1;
	elseif (x <= 6.3e+246)
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(y, i, z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+220], t$95$1, If[LessEqual[x, 6.3e+246], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000048e220 or 6.3e246 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if -6.00000000000000048e220 < x < 6.3e246

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+246}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (log y) x))))
   (if (<= x -6e+220)
     t_1
     (if (<= x 6.3e+246) (+ (fma (- b 0.5) (log c) (fma y i z)) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (log(y) * x));
	double tmp;
	if (x <= -6e+220) {
		tmp = t_1;
	} else if (x <= 6.3e+246) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (x <= -6e+220)
		tmp = t_1;
	elseif (x <= 6.3e+246)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + a);
	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[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+220], t$95$1, If[LessEqual[x, 6.3e+246], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000048e220 or 6.3e246 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if -6.00000000000000048e220 < x < 6.3e246

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.1%

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

Alternative 7: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;t\_1 + a\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b)))
   (if (<= (- b 0.5) -4e+222)
     (+ t_1 a)
     (if (<= (- b 0.5) 5e+170)
       (+ (fma (/ z a) a a) (* i y))
       (+ t_1 (+ a t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double tmp;
	if ((b - 0.5) <= -4e+222) {
		tmp = t_1 + a;
	} else if ((b - 0.5) <= 5e+170) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1 + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	tmp = 0.0
	if (Float64(b - 0.5) <= -4e+222)
		tmp = Float64(t_1 + a);
	elseif (Float64(b - 0.5) <= 5e+170)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = Float64(t_1 + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -4e+222], N[(t$95$1 + a), $MachinePrecision], If[LessEqual[N[(b - 0.5), $MachinePrecision], 5e+170], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -4.0000000000000002e222

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto b \cdot \log c + a \]
   10. Step-by-step derivation

   11. Applied rewrites 78.4%

       \[\leadsto b \cdot \log c + a \]

   if -4.0000000000000002e222 < (-.f64 b #s(literal 1/2 binary64)) < 4.99999999999999977e170

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.5%

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

   if 4.99999999999999977e170 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 76.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;\log c \cdot b + a\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log c \cdot b + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b + a\\ \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log c) b) a)))
   (if (<= (- b 0.5) -4e+222)
     t_1
     (if (<= (- b 0.5) 5e+170) (+ (fma (/ z a) a a) (* i y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (log(c) * b) + a;
	double tmp;
	if ((b - 0.5) <= -4e+222) {
		tmp = t_1;
	} else if ((b - 0.5) <= 5e+170) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(log(c) * b) + a)
	tmp = 0.0
	if (Float64(b - 0.5) <= -4e+222)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 5e+170)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -4e+222], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 5e+170], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -4.0000000000000002e222 or 4.99999999999999977e170 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in b around inf

      \[\leadsto b \cdot \log c + a \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto b \cdot \log c + a \]

   if -4.0000000000000002e222 < (-.f64 b #s(literal 1/2 binary64)) < 4.99999999999999977e170

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;\log c \cdot b + a\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log c \cdot b + a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b + i \cdot y\\ \mathbf{if}\;i \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 8.8 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (log c) b) (* i y))))
   (if (<= i -1.18e+111)
     t_1
     (if (<= i 8.8e+86) (+ (fma (- b 0.5) (log c) z) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (log(c) * b) + (i * y);
	double tmp;
	if (i <= -1.18e+111) {
		tmp = t_1;
	} else if (i <= 8.8e+86) {
		tmp = fma((b - 0.5), log(c), z) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(log(c) * b) + Float64(i * y))
	tmp = 0.0
	if (i <= -1.18e+111)
		tmp = t_1;
	elseif (i <= 8.8e+86)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.18e+111], t$95$1, If[LessEqual[i, 8.8e+86], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.1799999999999999e111 or 8.80000000000000013e86 < i

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.1799999999999999e111 < i < 8.80000000000000013e86

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 56.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;\log c \cdot b + i \cdot y\\ \mathbf{elif}\;i \leq 8.8 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;\log c \cdot b + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{if}\;i \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (/ z a) a a) (* i y))))
   (if (<= i -4.5e+108)
     t_1
     (if (<= i 2.7e+57) (+ (fma (- b 0.5) (log c) z) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma((z / a), a, a) + (i * y);
	double tmp;
	if (i <= -4.5e+108) {
		tmp = t_1;
	} else if (i <= 2.7e+57) {
		tmp = fma((b - 0.5), log(c), z) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(z / a), a, a) + Float64(i * y))
	tmp = 0.0
	if (i <= -4.5e+108)
		tmp = t_1;
	elseif (i <= 2.7e+57)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -4.5e+108], t$95$1, If[LessEqual[i, 2.7e+57], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -4.5e108 or 2.6999999999999998e57 < i

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 61.3%

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

   if -4.5e108 < i < 2.6999999999999998e57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 57.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b)))
   (if (<= (- b 0.5) -4e+222)
     t_1
     (if (<= (- b 0.5) 1e+183) (+ (fma (/ z a) a a) (* i y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double tmp;
	if ((b - 0.5) <= -4e+222) {
		tmp = t_1;
	} else if ((b - 0.5) <= 1e+183) {
		tmp = fma((z / a), a, a) + (i * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	tmp = 0.0
	if (Float64(b - 0.5) <= -4e+222)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 1e+183)
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -4e+222], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 1e+183], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -4.0000000000000002e222 or 9.99999999999999947e182 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -4.0000000000000002e222 < (-.f64 b #s(literal 1/2 binary64)) < 9.99999999999999947e182

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 77.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.0%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+222}:\\ \;\;\;\;\log c \cdot b\\ \mathbf{elif}\;b - 0.5 \leq 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log c \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 5.4e+102)
   (fma (- b 0.5) (log c) (fma y i z))
   (+ (fma (/ z a) a a) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 5.4e+102) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z));
	} else {
		tmp = fma((z / a), a, a) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 5.4e+102)
		tmp = fma(Float64(b - 0.5), log(c), fma(y, i, z));
	else
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 5.4e+102], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4000000000000002e102

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 61.3%

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

   if 5.4000000000000002e102 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 28.2

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 67.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.3e-54) (fma (/ (* i y) z) z z) (+ (fma (/ z a) a a) (* i y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.3e-54) {
		tmp = fma(((i * y) / z), z, z);
	} else {
		tmp = fma((z / a), a, a) + (i * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.3e-54)
		tmp = fma(Float64(Float64(i * y) / z), z, z);
	else
		tmp = Float64(fma(Float64(z / a), a, a) + Float64(i * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.3e-54], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.30000000000000001e-54

   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 z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 33.1%

      \[\leadsto \mathsf{fma}\left(\frac{y \cdot i}{z}, z, z\right) \]

   if 1.30000000000000001e-54 < a

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 33.7

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

   5. Applied rewrites 33.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 51.9%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right) + i \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -1.02e+155) (fma (/ a z) z z) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -1.02e+155) {
		tmp = fma((a / z), z, z);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -1.02e+155)
		tmp = fma(Float64(a / z), z, z);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -1.02e+155], N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02e155

   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 z around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 56.9%

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

   if -1.02e155 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -21000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= t -21000000000000.0) (fma (/ z t) t t) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (t <= -21000000000000.0) {
		tmp = fma((z / t), t, t);
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (t <= -21000000000000.0)
		tmp = fma(Float64(z / t), t, t);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[t, -21000000000000.0], N[(N[(z / t), $MachinePrecision] * t + t), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1e13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 44.0%

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

   if -2.1e13 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.8%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -21000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.9% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 85.8

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

5. Applied rewrites 85.8%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 53.9%

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

9. Final simplification 53.9%

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

Alternative 17: 38.1% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 85.8

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

5. Applied rewrites 85.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 67.6%

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

9. Taylor expanded in y around inf

   \[\leadsto i \cdot y + a \]
10. Step-by-step derivation

11. Applied rewrites 35.6%

    \[\leadsto y \cdot i + a \]

12. Final simplification 35.6%

    \[\leadsto i \cdot y + a \]
13. Add Preprocessing

Alternative 18: 23.9% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 24.4

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

5. Applied rewrites 24.4%

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

6. Final simplification 24.4%

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

Reproduce

herbie shell --seed 2024235 
(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)))