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

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

Alternatives: 16

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

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

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]

Derivation

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. Add Preprocessing

Alternative 2: 17.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;\frac{z}{y} \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{a}{y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* i y)
     (if (<= t_1 -50.0)
       (* (/ z y) y)
       (if (<= t_1 INFINITY) (* (/ a y) y) (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -50.0) {
		tmp = (z / y) * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (a / y) * y;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = i * y;
	} else if (t_1 <= -50.0) {
		tmp = (z / y) * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (a / y) * y;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = i * y
	elif t_1 <= -50.0:
		tmp = (z / y) * y
	elif t_1 <= math.inf:
		tmp = (a / y) * y
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -50.0)
		tmp = Float64(Float64(z / y) * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(a / y) * y);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = i * y;
	elseif (t_1 <= -50.0)
		tmp = (z / y) * y;
	elseif (t_1 <= Inf)
		tmp = (a / y) * y;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -50.0], N[(N[(z / y), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(a / y), $MachinePrecision] * y), $MachinePrecision], N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 94.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 9.7%

      \[\leadsto \frac{z}{y} \cdot y \]

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{t}{y} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 12.9%

      \[\leadsto \frac{t}{y} \cdot y \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{a}{y} \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 12.6%

       \[\leadsto \frac{a}{y} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 18.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+25} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (or (<= t_1 2e+25) (not (<= t_1 INFINITY))) (* i y) (* (/ a y) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if ((t_1 <= 2e+25) || !(t_1 <= ((double) INFINITY))) {
		tmp = i * y;
	} else {
		tmp = (a / y) * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if ((t_1 <= 2e+25) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = i * y;
	} else {
		tmp = (a / y) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if (t_1 <= 2e+25) or not (t_1 <= math.inf):
		tmp = i * y
	else:
		tmp = (a / y) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if ((t_1 <= 2e+25) || !(t_1 <= Inf))
		tmp = Float64(i * y);
	else
		tmp = Float64(Float64(a / y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if ((t_1 <= 2e+25) || ~((t_1 <= Inf)))
		tmp = i * y;
	else
		tmp = (a / y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, 2e+25], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(i * y), $MachinePrecision], N[(N[(a / y), $MachinePrecision] * y), $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)) < 2.00000000000000018e25 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 26.9

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

   5. Applied rewrites 26.9%

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

   if 2.00000000000000018e25 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{t}{y} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 13.1%

      \[\leadsto \frac{t}{y} \cdot y \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{a}{y} \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 12.7%

       \[\leadsto \frac{a}{y} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+25} \lor \neg \left(\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq \infty\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, t\_1\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log y) x (fma (- b 0.5) (log c) t))))
   (if (<=
        (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
        2e+25)
     (+ (fma i y z) t_1)
     (+ (fma i y t_1) a))))

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((b - 0.5), log(c), t));
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 2e+25) {
		tmp = fma(i, y, z) + t_1;
	} else {
		tmp = fma(i, y, t_1) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(y), x, fma(Float64(b - 0.5), log(c), t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 2e+25)
		tmp = Float64(fma(i, y, z) + t_1);
	else
		tmp = Float64(fma(i, y, t_1) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 2e+25], N[(N[(i * y + z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(i * y + t$95$1), $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)) < 2.00000000000000018e25

   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 a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if 2.00000000000000018e25 < (+.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 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 84.2%

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

Alternative 5: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, i \cdot y\right) + \left(t + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      2e+25)
   (+ (fma (- b 0.5) (log c) (* i y)) (+ t z))
   (+ (fma i y (fma (log c) (- b 0.5) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 2e+25) {
		tmp = fma((b - 0.5), log(c), (i * y)) + (t + z);
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 2e+25)
		tmp = Float64(fma(Float64(b - 0.5), log(c), Float64(i * y)) + Float64(t + z));
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], 2e+25], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision] + N[(t + z), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $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)) < 2.00000000000000018e25

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.9%

      \[\leadsto \left(\left(t + \color{blue}{z}\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if 2.00000000000000018e25 < (+.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 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

Alternative 6: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b - 0.5, \log c, t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      2e+25)
   (fma y i (fma (- b 0.5) (log c) (+ t z)))
   (+ (fma i y (fma (log c) (- b 0.5) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 2e+25) {
		tmp = fma(y, i, fma((b - 0.5), log(c), (t + z)));
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 2e+25)
		tmp = fma(y, i, fma(Float64(b - 0.5), log(c), Float64(t + z)));
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], 2e+25], N[(y * i + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $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)) < 2.00000000000000018e25

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.9%

      \[\leadsto \left(\left(t + \color{blue}{z}\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if 2.00000000000000018e25 < (+.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 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

Alternative 7: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, t\right)\right)\right) + a\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -1.35e+188)
   (+ (fma i y (fma (log y) x (fma -0.5 (log c) t))) a)
   (if (<= x 1.6e+168)
     (+ (+ (+ (+ t z) a) (* (- b 0.5) (log c))) (* y i))
     (+ (fma i y z) (fma (log y) x (fma (- b 0.5) (log c) t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -1.35e+188) {
		tmp = fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a;
	} else if (x <= 1.6e+168) {
		tmp = (((t + z) + a) + ((b - 0.5) * log(c))) + (y * i);
	} else {
		tmp = fma(i, y, z) + fma(log(y), x, fma((b - 0.5), log(c), t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -1.35e+188)
		tmp = Float64(fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a);
	elseif (x <= 1.6e+168)
		tmp = Float64(Float64(Float64(Float64(t + z) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	else
		tmp = Float64(fma(i, y, z) + fma(log(y), x, fma(Float64(b - 0.5), log(c), t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -1.35e+188], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(-0.5 * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 1.6e+168], N[(N[(N[(N[(t + z), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e188

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 90.3%

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

   if -1.35e188 < x < 1.6000000000000001e168

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 1.6000000000000001e168 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 89.2

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

   5. Applied rewrites 89.2%

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

Alternative 8: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+188} \lor \neg \left(x \leq 5.2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, t\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.35e+188) (not (<= x 5.2e+178)))
   (+ (fma i y (fma (log y) x (fma -0.5 (log c) t))) a)
   (+ (+ (+ (+ t z) 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) {
	double tmp;
	if ((x <= -1.35e+188) || !(x <= 5.2e+178)) {
		tmp = fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a;
	} else {
		tmp = (((t + z) + a) + ((b - 0.5) * log(c))) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.35e+188) || !(x <= 5.2e+178))
		tmp = Float64(fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a);
	else
		tmp = Float64(Float64(Float64(Float64(t + z) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.35e+188], N[Not[LessEqual[x, 5.2e+178]], $MachinePrecision]], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(-0.5 * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(N[(t + z), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e188 or 5.2000000000000001e178 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 89.1%

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

   if -1.35e188 < x < 5.2000000000000001e178

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+188} \lor \neg \left(x \leq 5.2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, t\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+219} \lor \neg \left(x \leq 1.45 \cdot 10^{+180}\right):\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(\log c, -0.5 + b, \log y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -3.7e+219) (not (<= x 1.45e+180)))
   (+ (+ a t) (fma (log c) (+ -0.5 b) (* (log y) x)))
   (+ (+ (+ (+ t z) 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) {
	double tmp;
	if ((x <= -3.7e+219) || !(x <= 1.45e+180)) {
		tmp = (a + t) + fma(log(c), (-0.5 + b), (log(y) * x));
	} else {
		tmp = (((t + z) + a) + ((b - 0.5) * log(c))) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -3.7e+219) || !(x <= 1.45e+180))
		tmp = Float64(Float64(a + t) + fma(log(c), Float64(-0.5 + b), Float64(log(y) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(t + z) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -3.7e+219], N[Not[LessEqual[x, 1.45e+180]], $MachinePrecision]], N[(N[(a + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(-0.5 + b), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t + z), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e219 or 1.45000000000000004e180 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.8%

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

   if -3.7e219 < x < 1.45000000000000004e180

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+219} \lor \neg \left(x \leq 1.45 \cdot 10^{+180}\right):\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(\log c, -0.5 + b, \log y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+223} \lor \neg \left(x \leq 1.8 \cdot 10^{+200}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -8.8e+223) (not (<= x 1.8e+200)))
   (* (log y) x)
   (+ (+ (+ (+ t z) 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) {
	double tmp;
	if ((x <= -8.8e+223) || !(x <= 1.8e+200)) {
		tmp = log(y) * x;
	} else {
		tmp = (((t + z) + a) + ((b - 0.5) * log(c))) + (y * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((x <= (-8.8d+223)) .or. (.not. (x <= 1.8d+200))) then
        tmp = log(y) * x
    else
        tmp = (((t + z) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -8.8e+223) || !(x <= 1.8e+200)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (((t + z) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -8.8e+223) or not (x <= 1.8e+200):
		tmp = math.log(y) * x
	else:
		tmp = (((t + z) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -8.8e+223) || !(x <= 1.8e+200))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(t + z) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -8.8e+223) || ~((x <= 1.8e+200)))
		tmp = log(y) * x;
	else
		tmp = (((t + z) + a) + ((b - 0.5) * log(c))) + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -8.8e+223], N[Not[LessEqual[x, 1.8e+200]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(t + z), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999999e223 or 1.7999999999999999e200 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 24.3

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

   5. Applied rewrites 24.3%

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

   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 76.0

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

   8. Applied rewrites 76.0%

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

   if -8.7999999999999999e223 < x < 1.7999999999999999e200

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+223} \lor \neg \left(x \leq 1.8 \cdot 10^{+200}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + z\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+223} \lor \neg \left(x \leq 1.8 \cdot 10^{+200}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -8.8e+223) (not (<= x 1.8e+200)))
   (* (log y) x)
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -8.8e+223) || !(x <= 1.8e+200)) {
		tmp = log(y) * x;
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -8.8e+223) || !(x <= 1.8e+200))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -8.8e+223], N[Not[LessEqual[x, 1.8e+200]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999999e223 or 1.7999999999999999e200 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 24.3

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

   5. Applied rewrites 24.3%

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

   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 76.0

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

   8. Applied rewrites 76.0%

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

   if -8.7999999999999999e223 < x < 1.7999999999999999e200

   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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+223} \lor \neg \left(x \leq 1.8 \cdot 10^{+200}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+221} \lor \neg \left(x \leq 3.5 \cdot 10^{+176}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -4.2e+221) (not (<= x 3.5e+176)))
   (* (log y) x)
   (+ (fma i y (fma (log c) (- b 0.5) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -4.2e+221) || !(x <= 3.5e+176)) {
		tmp = log(y) * x;
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -4.2e+221) || !(x <= 3.5e+176))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -4.2e+221], N[Not[LessEqual[x, 3.5e+176]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000004e221 or 3.50000000000000003e176 < x

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 30.6

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

   5. Applied rewrites 30.6%

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

   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 69.0

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

   8. Applied rewrites 69.0%

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

   if -4.20000000000000004e221 < x < 3.50000000000000003e176

   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 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+221} \lor \neg \left(x \leq 3.5 \cdot 10^{+176}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, \left(t + z\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -5.8e+38)
   (fma y i (fma (log c) -0.5 (+ (+ t z) a)))
   (+ (fma i y (fma (log c) (- b 0.5) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -5.8e+38) {
		tmp = fma(y, i, fma(log(c), -0.5, ((t + z) + a)));
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -5.8e+38)
		tmp = fma(y, i, fma(log(c), -0.5, Float64(Float64(t + z) + a)));
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -5.8e+38], N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + N[(N[(t + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000013e38

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \left(\color{blue}{\left(\left(t + z\right) + a\right)} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 90.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 86.1%

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

   if -5.80000000000000013e38 < z

   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 z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

Alternative 14: 38.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.7 \cdot 10^{+72} \lor \neg \left(i \leq 1.9 \cdot 10^{+65}\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= i -2.7e+72) (not (<= i 1.9e+65))) (* i y) (fma (/ z a) a a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i <= -2.7e+72) || !(i <= 1.9e+65)) {
		tmp = i * y;
	} else {
		tmp = fma((z / a), a, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((i <= -2.7e+72) || !(i <= 1.9e+65))
		tmp = Float64(i * y);
	else
		tmp = fma(Float64(z / a), a, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[i, -2.7e+72], N[Not[LessEqual[i, 1.9e+65]], $MachinePrecision]], N[(i * y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.7000000000000001e72 or 1.90000000000000006e65 < i

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   if -2.7000000000000001e72 < i < 1.90000000000000006e65

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 31.3%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.7 \cdot 10^{+72} \lor \neg \left(i \leq 1.9 \cdot 10^{+65}\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{a}, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -2.7e+144) (fma (/ z a) a a) (fma (/ (* i y) a) a a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2.7e+144) {
		tmp = fma((z / a), a, a);
	} else {
		tmp = fma(((i * y) / a), a, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2.7e+144)
		tmp = fma(Float64(z / a), a, a);
	else
		tmp = fma(Float64(Float64(i * y) / a), a, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2.7e+144], N[(N[(z / a), $MachinePrecision] * a + a), $MachinePrecision], N[(N[(N[(i * y), $MachinePrecision] / a), $MachinePrecision] * a + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000015e144

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 36.3%

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

   if -2.70000000000000015e144 < z

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 40.0%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{a}, a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.1% 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 99.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 27.3

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

5. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2024313 
(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)))