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

Percentage Accurate: 99.8% → 99.8%

Time: 10.6s

Alternatives: 13

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \log c \cdot b\right)\right) + a\\ \mathbf{elif}\;t\_1 \leq 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, z\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) (- b 0.5))))
   (if (<= t_1 -2e+145)
     (+ (fma i y (fma (log y) x (* (log c) b))) a)
     (if (<= t_1 1e+171)
       (+ (fma i y (fma (log y) x (fma -0.5 (log c) z))) a)
       (+ (fma (log y) x (fma (log c) (- b 0.5) z)) a)))))

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 - 0.5);
	double tmp;
	if (t_1 <= -2e+145) {
		tmp = fma(i, y, fma(log(y), x, (log(c) * b))) + a;
	} else if (t_1 <= 1e+171) {
		tmp = fma(i, y, fma(log(y), x, fma(-0.5, log(c), z))) + a;
	} else {
		tmp = fma(log(y), x, fma(log(c), (b - 0.5), z)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * Float64(b - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+145)
		tmp = Float64(fma(i, y, fma(log(y), x, Float64(log(c) * b))) + a);
	elseif (t_1 <= 1e+171)
		tmp = Float64(fma(i, y, fma(log(y), x, fma(-0.5, log(c), z))) + a);
	else
		tmp = Float64(fma(log(y), x, fma(log(c), Float64(b - 0.5), z)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+145], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$1, 1e+171], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(-0.5 * N[Log[c], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2e145

   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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 47.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 70.5%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 90.1%

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

   if -2e145 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.99999999999999954e170

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 79.8%

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

   if 9.99999999999999954e170 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log c \cdot \left(b - 0.5\right) \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \log c \cdot b\right)\right) + a\\ \mathbf{elif}\;\log c \cdot \left(b - 0.5\right) \leq 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, z\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{t}{z}, z, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* i y) (+ (* (log c) (- b 0.5)) (+ a (+ t (+ z (* (log y) x))))))
      -2e+115)
   (fma y i (fma (/ t z) z z))
   (fma y i (fma (/ a z) z z))))

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)))))) <= -2e+115) {
		tmp = fma(y, i, fma((t / z), z, z));
	} else {
		tmp = fma(y, i, fma((a / z), z, z));
	}
	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)))))) <= -2e+115)
		tmp = fma(y, i, fma(Float64(t / z), z, z));
	else
		tmp = fma(y, i, fma(Float64(a / z), z, z));
	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], -2e+115], N[(y * i + N[(N[(t / z), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2e115

   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 z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 43.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 43.8

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

   10. Applied rewrites 43.8%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 45.2%

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

   if -2e115 < (+.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 99.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 41.2

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

   10. Applied rewrites 41.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot y + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right)\right) \leq -2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{t}{z}, z, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \log c \cdot b\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -6.6e+137)
   (+ (fma (log y) x (fma (log c) (- b 0.5) z)) a)
   (if (<= x 4.1e+82)
     (+ (fma (- b 0.5) (log c) (fma i y z)) (+ a t))
     (+ (fma i y (fma (log y) x (* (log c) b))) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -6.6e+137) {
		tmp = fma(log(y), x, fma(log(c), (b - 0.5), z)) + a;
	} else if (x <= 4.1e+82) {
		tmp = fma((b - 0.5), log(c), fma(i, y, z)) + (a + t);
	} else {
		tmp = fma(i, y, fma(log(y), x, (log(c) * b))) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -6.6e+137)
		tmp = Float64(fma(log(y), x, fma(log(c), Float64(b - 0.5), z)) + a);
	elseif (x <= 4.1e+82)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, z)) + Float64(a + t));
	else
		tmp = Float64(fma(i, y, fma(log(y), x, Float64(log(c) * b))) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -6.6e+137], N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 4.1e+82], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.60000000000000005e137

   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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.8%

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

   if -6.60000000000000005e137 < x < 4.09999999999999995e82

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 4.09999999999999995e82 < 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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 74.4%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 77.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \log c \cdot b\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (log y) x (fma (log c) (- b 0.5) z)) a)))
   (if (<= x -6.6e+137)
     t_1
     (if (<= x 2.15e+102)
       (+ (fma (- b 0.5) (log c) (fma i y z)) (+ a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(y), x, fma(log(c), (b - 0.5), z)) + a;
	double tmp;
	if (x <= -6.6e+137) {
		tmp = t_1;
	} else if (x <= 2.15e+102) {
		tmp = fma((b - 0.5), log(c), fma(i, y, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(log(y), x, fma(log(c), Float64(b - 0.5), z)) + a)
	tmp = 0.0
	if (x <= -6.6e+137)
		tmp = t_1;
	elseif (x <= 2.15e+102)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[x, -6.6e+137], t$95$1, If[LessEqual[x, 2.15e+102], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.60000000000000005e137 or 2.15e102 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 78.0%

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

   if -6.60000000000000005e137 < x < 2.15e102

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 83.4%

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

Alternative 7: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.9 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{t\_1}{z}, z, z\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.9e+174)
     (fma y i (fma (/ t_1 z) z z))
     (if (<= x 2.9e+256)
       (+ (fma (- b 0.5) (log c) (fma i y z)) (+ a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.9e+174) {
		tmp = fma(y, i, fma((t_1 / z), z, z));
	} else if (x <= 2.9e+256) {
		tmp = fma((b - 0.5), log(c), fma(i, y, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.9e+174)
		tmp = fma(y, i, fma(Float64(t_1 / z), z, z));
	elseif (x <= 2.9e+256)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.9e+174], N[(y * i + N[(N[(t$95$1 / z), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+256], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.9000000000000002e174

   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 z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 16.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 16.9

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

   10. Applied rewrites 16.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 54.0%

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

   if -6.9000000000000002e174 < x < 2.9000000000000002e256

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if 2.9000000000000002e256 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.1

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

   8. Applied rewrites 83.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, z\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4 \cdot 10^{+176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4e+176)
     t_1
     (if (<= x 2.9e+256)
       (+ (fma (- b 0.5) (log c) (fma i y z)) (+ a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -4e+176) {
		tmp = t_1;
	} else if (x <= 2.9e+256) {
		tmp = fma((b - 0.5), log(c), fma(i, y, z)) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -4e+176)
		tmp = t_1;
	elseif (x <= 2.9e+256)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(i, y, z)) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4e+176], t$95$1, If[LessEqual[x, 2.9e+256], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e176 or 2.9000000000000002e256 < 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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4e176 < x < 2.9000000000000002e256

   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. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+176}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4 \cdot 10^{+176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4e+176)
     t_1
     (if (<= x 2.9e+256) (+ (fma i y (fma (log c) (- b 0.5) 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(y) * x;
	double tmp;
	if (x <= -4e+176) {
		tmp = t_1;
	} else if (x <= 2.9e+256) {
		tmp = fma(i, y, fma(log(c), (b - 0.5), z)) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -4e+176)
		tmp = t_1;
	elseif (x <= 2.9e+256)
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), 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[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4e+176], t$95$1, If[LessEqual[x, 2.9e+256], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e176 or 2.9000000000000002e256 < 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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4e176 < x < 2.9000000000000002e256

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.7%

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

Alternative 10: 50.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b)))
   (if (<= (- b 0.5) -4e+153)
     t_1
     (if (<= (- b 0.5) 5e+176) (fma y i (fma (/ a z) z 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 = log(c) * b;
	double tmp;
	if ((b - 0.5) <= -4e+153) {
		tmp = t_1;
	} else if ((b - 0.5) <= 5e+176) {
		tmp = fma(y, i, fma((a / z), z, z));
	} 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+153)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 5e+176)
		tmp = fma(y, i, fma(Float64(a / z), z, z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -4e+153], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 5e+176], N[(y * i + N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -4e153 or 5e176 < (-.f64 b #s(literal 1/2 binary64))

   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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -4e153 < (-.f64 b #s(literal 1/2 binary64)) < 5e176

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 49.3

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

   10. Applied rewrites 49.3%

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

Alternative 11: 50.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -3.85 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -3.85e+174)
     t_1
     (if (<= x 2.2e+163) (fma y i (fma (/ a z) z 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 = log(y) * x;
	double tmp;
	if (x <= -3.85e+174) {
		tmp = t_1;
	} else if (x <= 2.2e+163) {
		tmp = fma(y, i, fma((a / z), z, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -3.85e+174)
		tmp = t_1;
	elseif (x <= 2.2e+163)
		tmp = fma(y, i, fma(Float64(a / z), z, z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.85e+174], t$95$1, If[LessEqual[x, 2.2e+163], N[(y * i + N[(N[(a / z), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.84999999999999979e174 or 2.19999999999999986e163 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 67.2

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

   8. Applied rewrites 67.2%

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

   if -3.84999999999999979e174 < x < 2.19999999999999986e163

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 49.3

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

   10. Applied rewrites 49.3%

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

Alternative 12: 44.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 42.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 42.4

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

10. Applied rewrites 42.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\frac{a}{z}, z, z\right)\right)} \]
11. Add Preprocessing

Alternative 13: 24.1% 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 99.8%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 22.2

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

5. Applied rewrites 22.2%

   \[\leadsto \color{blue}{i \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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)))