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

Percentage Accurate: 99.8% → 99.8%

Time: 25.5s

Alternatives: 21

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+l+ 99.9%

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

   8. +-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. associate-+l+ 99.9%

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

   11. fma-def 99.9%

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

   12. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ a (+ t (+ z (* x (log y))))) (* (- 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 ((a + (t + (z + (x * log(y))))) + ((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 = ((a + (t + (z + (x * log(y))))) + ((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 ((a + (t + (z + (x * Math.log(y))))) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{+50} \lor \neg \left(i \leq 4.5 \cdot 10^{-63}\right):\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + x \cdot \log y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= i -1.2e+50) (not (<= i 4.5e-63)))
   (+ (* y i) (+ (* b (log c)) (+ a (+ z t))))
   (+ (* (- b 0.5) (log c)) (+ a (+ z (* x (log y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i <= -1.2e+50) || !(i <= 4.5e-63)) {
		tmp = (y * i) + ((b * log(c)) + (a + (z + t)));
	} else {
		tmp = ((b - 0.5) * log(c)) + (a + (z + (x * log(y))));
	}
	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 ((i <= (-1.2d+50)) .or. (.not. (i <= 4.5d-63))) then
        tmp = (y * i) + ((b * log(c)) + (a + (z + t)))
    else
        tmp = ((b - 0.5d0) * log(c)) + (a + (z + (x * log(y))))
    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 ((i <= -1.2e+50) || !(i <= 4.5e-63)) {
		tmp = (y * i) + ((b * Math.log(c)) + (a + (z + t)));
	} else {
		tmp = ((b - 0.5) * Math.log(c)) + (a + (z + (x * Math.log(y))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (i <= -1.2e+50) or not (i <= 4.5e-63):
		tmp = (y * i) + ((b * math.log(c)) + (a + (z + t)))
	else:
		tmp = ((b - 0.5) * math.log(c)) + (a + (z + (x * math.log(y))))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((i <= -1.2e+50) || !(i <= 4.5e-63))
		tmp = Float64(Float64(y * i) + Float64(Float64(b * log(c)) + Float64(a + Float64(z + t))));
	else
		tmp = Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(a + Float64(z + Float64(x * log(y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((i <= -1.2e+50) || ~((i <= 4.5e-63)))
		tmp = (y * i) + ((b * log(c)) + (a + (z + t)));
	else
		tmp = ((b - 0.5) * log(c)) + (a + (z + (x * log(y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[i, -1.2e+50], N[Not[LessEqual[i, 4.5e-63]], $MachinePrecision]], N[(N[(y * i), $MachinePrecision] + N[(N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.2000000000000001e50 or 4.5e-63 < i

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 91.0%

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

   3. Taylor expanded in b around inf 91.0%

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

   if -1.2000000000000001e50 < i < 4.5e-63

   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. Taylor expanded in t around 0 81.5%

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

   3. Taylor expanded in y around 0 80.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{+50} \lor \neg \left(i \leq 4.5 \cdot 10^{-63}\right):\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + x \cdot \log y\right)\right)\\ \end{array} \]

Alternative 4: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;i \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-64}:\\ \;\;\;\;t_1 + \left(a + \left(z + x \cdot \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(t_1 + \left(z + t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= i -1.15e+50)
     (+ (* y i) (+ (* b (log c)) (+ a (+ z t))))
     (if (<= i 2e-64)
       (+ t_1 (+ a (+ z (* x (log y)))))
       (fma y i (+ a (+ t_1 (+ z t))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (i <= -1.15e+50) {
		tmp = (y * i) + ((b * log(c)) + (a + (z + t)));
	} else if (i <= 2e-64) {
		tmp = t_1 + (a + (z + (x * log(y))));
	} else {
		tmp = fma(y, i, (a + (t_1 + (z + t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (i <= -1.15e+50)
		tmp = Float64(Float64(y * i) + Float64(Float64(b * log(c)) + Float64(a + Float64(z + t))));
	elseif (i <= 2e-64)
		tmp = Float64(t_1 + Float64(a + Float64(z + Float64(x * log(y)))));
	else
		tmp = fma(y, i, Float64(a + Float64(t_1 + Float64(z + t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.15e+50], N[(N[(y * i), $MachinePrecision] + N[(N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2e-64], N[(t$95$1 + N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(a + N[(t$95$1 + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.14999999999999998e50

   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. Taylor expanded in x around 0 92.4%

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

   3. Taylor expanded in b around inf 92.4%

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

   if -1.14999999999999998e50 < i < 1.99999999999999993e-64

   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. Taylor expanded in t around 0 81.5%

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

   3. Taylor expanded in y around 0 80.8%

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

   if 1.99999999999999993e-64 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + x \cdot \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(\left(b - 0.5\right) \cdot \log c + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 5: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (y * i) + (((b - 0.5) * log(c)) + (a + (z + (x * log(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 = (y * i) + (((b - 0.5d0) * log(c)) + (a + (z + (x * log(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 (y * i) + (((b - 0.5) * Math.log(c)) + (a + (z + (x * Math.log(y)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in t around 0 84.4%

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

3. Final simplification 84.4%

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

Alternative 6: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+202} \lor \neg \left(x \leq 8.2 \cdot 10^{+211}\right):\\ \;\;\;\;\left(a + \left(z + x \cdot \log y\right)\right) + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -7.5e+202) (not (<= x 8.2e+211)))
   (+ (+ a (+ z (* x (log y)))) (* b (log c)))
   (+ (* y i) (+ (* (- b 0.5) (log c)) (+ a (+ z t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -7.5e+202) || !(x <= 8.2e+211)) {
		tmp = (a + (z + (x * log(y)))) + (b * log(c));
	} else {
		tmp = (y * i) + (((b - 0.5) * log(c)) + (a + (z + t)));
	}
	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 <= (-7.5d+202)) .or. (.not. (x <= 8.2d+211))) then
        tmp = (a + (z + (x * log(y)))) + (b * log(c))
    else
        tmp = (y * i) + (((b - 0.5d0) * log(c)) + (a + (z + t)))
    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 <= -7.5e+202) || !(x <= 8.2e+211)) {
		tmp = (a + (z + (x * Math.log(y)))) + (b * Math.log(c));
	} else {
		tmp = (y * i) + (((b - 0.5) * Math.log(c)) + (a + (z + t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -7.5e+202) or not (x <= 8.2e+211):
		tmp = (a + (z + (x * math.log(y)))) + (b * math.log(c))
	else:
		tmp = (y * i) + (((b - 0.5) * math.log(c)) + (a + (z + t)))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -7.5e+202) || !(x <= 8.2e+211))
		tmp = Float64(Float64(a + Float64(z + Float64(x * log(y)))) + Float64(b * log(c)));
	else
		tmp = Float64(Float64(y * i) + Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(a + Float64(z + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -7.5e+202) || ~((x <= 8.2e+211)))
		tmp = (a + (z + (x * log(y)))) + (b * log(c));
	else
		tmp = (y * i) + (((b - 0.5) * log(c)) + (a + (z + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -7.5e+202], N[Not[LessEqual[x, 8.2e+211]], $MachinePrecision]], N[(N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4999999999999999e202 or 8.1999999999999998e211 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in t around 0 97.2%

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

   3. Taylor expanded in y around 0 90.4%

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

   4. Taylor expanded in b around inf 90.4%

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

   if -7.4999999999999999e202 < x < 8.1999999999999998e211

   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. Taylor expanded in x around 0 94.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+202} \lor \neg \left(x \leq 8.2 \cdot 10^{+211}\right):\\ \;\;\;\;\left(a + \left(z + x \cdot \log y\right)\right) + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+222}:\\ \;\;\;\;t_2 + t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+218}:\\ \;\;\;\;y \cdot i + \left(t_2 + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (* (- b 0.5) (log c))))
   (if (<= x -1.45e+222)
     (+ t_2 t_1)
     (if (<= x 2.8e+218) (+ (* y i) (+ t_2 (+ a (+ z t)))) (+ a (+ z t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (x <= -1.45e+222) {
		tmp = t_2 + t_1;
	} else if (x <= 2.8e+218) {
		tmp = (y * i) + (t_2 + (a + (z + t)));
	} else {
		tmp = a + (z + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = (b - 0.5d0) * log(c)
    if (x <= (-1.45d+222)) then
        tmp = t_2 + t_1
    else if (x <= 2.8d+218) then
        tmp = (y * i) + (t_2 + (a + (z + t)))
    else
        tmp = a + (z + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * Math.log(y);
	double t_2 = (b - 0.5) * Math.log(c);
	double tmp;
	if (x <= -1.45e+222) {
		tmp = t_2 + t_1;
	} else if (x <= 2.8e+218) {
		tmp = (y * i) + (t_2 + (a + (z + t)));
	} else {
		tmp = a + (z + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x * math.log(y)
	t_2 = (b - 0.5) * math.log(c)
	tmp = 0
	if x <= -1.45e+222:
		tmp = t_2 + t_1
	elif x <= 2.8e+218:
		tmp = (y * i) + (t_2 + (a + (z + t)))
	else:
		tmp = a + (z + t_1)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (x <= -1.45e+222)
		tmp = Float64(t_2 + t_1);
	elseif (x <= 2.8e+218)
		tmp = Float64(Float64(y * i) + Float64(t_2 + Float64(a + Float64(z + t))));
	else
		tmp = Float64(a + Float64(z + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x * log(y);
	t_2 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (x <= -1.45e+222)
		tmp = t_2 + t_1;
	elseif (x <= 2.8e+218)
		tmp = (y * i) + (t_2 + (a + (z + t)));
	else
		tmp = a + (z + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+222], N[(t$95$2 + t$95$1), $MachinePrecision], If[LessEqual[x, 2.8e+218], N[(N[(y * i), $MachinePrecision] + N[(t$95$2 + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999991e222

   1. Initial program 99.7%

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

   2. Taylor expanded in t around 0 93.5%

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

   3. Taylor expanded in y around 0 83.5%

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

   4. Taylor expanded in a around 0 83.5%

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

   5. Taylor expanded in x around inf 82.3%

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

   if -1.44999999999999991e222 < x < 2.7999999999999998e218

   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. Taylor expanded in x around 0 94.5%

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

   if 2.7999999999999998e218 < 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. Taylor expanded in t around 0 99.8%

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

   3. Taylor expanded in y around 0 95.3%

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

   4. Taylor expanded in b around inf 95.3%

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

   5. Taylor expanded in b around 0 90.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+222}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + x \cdot \log y\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+218}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \end{array} \]

Alternative 8: 58.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + b \cdot \log c\right)\\ t_2 := \mathsf{fma}\left(y, i, z + a\right)\\ t_3 := a + \left(z + x \cdot \log y\right)\\ \mathbf{if}\;i \leq -4 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -4.7 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.45 \cdot 10^{-302}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 0.00012:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (* b (log c)))))
        (t_2 (fma y i (+ z a)))
        (t_3 (+ a (+ z (* x (log y))))))
   (if (<= i -4e+59)
     t_2
     (if (<= i -4.7e-182)
       t_1
       (if (<= i 1.45e-302)
         t_3
         (if (<= i 3.4e-195)
           t_1
           (if (<= i 9e-90) t_3 (if (<= i 0.00012) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (z + (b * log(c)));
	double t_2 = fma(y, i, (z + a));
	double t_3 = a + (z + (x * log(y)));
	double tmp;
	if (i <= -4e+59) {
		tmp = t_2;
	} else if (i <= -4.7e-182) {
		tmp = t_1;
	} else if (i <= 1.45e-302) {
		tmp = t_3;
	} else if (i <= 3.4e-195) {
		tmp = t_1;
	} else if (i <= 9e-90) {
		tmp = t_3;
	} else if (i <= 0.00012) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(z + Float64(b * log(c))))
	t_2 = fma(y, i, Float64(z + a))
	t_3 = Float64(a + Float64(z + Float64(x * log(y))))
	tmp = 0.0
	if (i <= -4e+59)
		tmp = t_2;
	elseif (i <= -4.7e-182)
		tmp = t_1;
	elseif (i <= 1.45e-302)
		tmp = t_3;
	elseif (i <= 3.4e-195)
		tmp = t_1;
	elseif (i <= 9e-90)
		tmp = t_3;
	elseif (i <= 0.00012)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(z + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -4e+59], t$95$2, If[LessEqual[i, -4.7e-182], t$95$1, If[LessEqual[i, 1.45e-302], t$95$3, If[LessEqual[i, 3.4e-195], t$95$1, If[LessEqual[i, 9e-90], t$95$3, If[LessEqual[i, 0.00012], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.99999999999999989e59 or 1.20000000000000003e-4 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.5%

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

   5. Taylor expanded in z around inf 70.9%

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

   if -3.99999999999999989e59 < i < -4.7e-182 or 1.44999999999999997e-302 < i < 3.40000000000000001e-195 or 9.00000000000000017e-90 < i < 1.20000000000000003e-4

   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. Taylor expanded in t around 0 81.9%

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

   3. Taylor expanded in y around 0 78.0%

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

   4. Taylor expanded in b around inf 76.4%

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

   5. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{a + \left(\log c \cdot b + z\right)} \]

   if -4.7e-182 < i < 1.44999999999999997e-302 or 3.40000000000000001e-195 < i < 9.00000000000000017e-90

   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. Taylor expanded in t around 0 82.6%

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

   3. Taylor expanded in y around 0 82.6%

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

   4. Taylor expanded in b around inf 78.8%

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

   5. Taylor expanded in b around 0 77.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{elif}\;i \leq -4.7 \cdot 10^{-182}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{elif}\;i \leq 1.45 \cdot 10^{-302}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{-195}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-90}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 0.00012:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \]

Alternative 9: 60.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := a + \left(z + t_1\right)\\ t_3 := \mathsf{fma}\left(y, i, z + a\right)\\ t_4 := a + \left(z + x \cdot \log y\right)\\ \mathbf{if}\;i \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -3.5 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 5.1 \cdot 10^{-304}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{-195}:\\ \;\;\;\;t_1 + \left(a + \left(z + t\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 0.000125:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c)))
        (t_2 (+ a (+ z t_1)))
        (t_3 (fma y i (+ z a)))
        (t_4 (+ a (+ z (* x (log y))))))
   (if (<= i -6.2e+59)
     t_3
     (if (<= i -3.5e-183)
       t_2
       (if (<= i 5.1e-304)
         t_4
         (if (<= i 5.5e-195)
           (+ t_1 (+ a (+ z t)))
           (if (<= i 5.5e-90) t_4 (if (<= i 0.000125) t_2 t_3))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = a + (z + t_1);
	double t_3 = fma(y, i, (z + a));
	double t_4 = a + (z + (x * log(y)));
	double tmp;
	if (i <= -6.2e+59) {
		tmp = t_3;
	} else if (i <= -3.5e-183) {
		tmp = t_2;
	} else if (i <= 5.1e-304) {
		tmp = t_4;
	} else if (i <= 5.5e-195) {
		tmp = t_1 + (a + (z + t));
	} else if (i <= 5.5e-90) {
		tmp = t_4;
	} else if (i <= 0.000125) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(a + Float64(z + t_1))
	t_3 = fma(y, i, Float64(z + a))
	t_4 = Float64(a + Float64(z + Float64(x * log(y))))
	tmp = 0.0
	if (i <= -6.2e+59)
		tmp = t_3;
	elseif (i <= -3.5e-183)
		tmp = t_2;
	elseif (i <= 5.1e-304)
		tmp = t_4;
	elseif (i <= 5.5e-195)
		tmp = Float64(t_1 + Float64(a + Float64(z + t)));
	elseif (i <= 5.5e-90)
		tmp = t_4;
	elseif (i <= 0.000125)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -6.2e+59], t$95$3, If[LessEqual[i, -3.5e-183], t$95$2, If[LessEqual[i, 5.1e-304], t$95$4, If[LessEqual[i, 5.5e-195], N[(t$95$1 + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.5e-90], t$95$4, If[LessEqual[i, 0.000125], t$95$2, t$95$3]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -6.20000000000000029e59 or 1.25e-4 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.5%

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

   5. Taylor expanded in z around inf 70.9%

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

   if -6.20000000000000029e59 < i < -3.49999999999999991e-183 or 5.5000000000000003e-90 < i < 1.25e-4

   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. Taylor expanded in t around 0 79.7%

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

   3. Taylor expanded in y around 0 74.7%

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

   4. Taylor expanded in b around inf 74.7%

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

   5. Taylor expanded in x around 0 64.0%

      \[\leadsto \color{blue}{a + \left(\log c \cdot b + z\right)} \]

   if -3.49999999999999991e-183 < i < 5.09999999999999979e-304 or 5.5000000000000003e-195 < i < 5.5000000000000003e-90

   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. Taylor expanded in t around 0 82.6%

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

   3. Taylor expanded in y around 0 82.6%

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

   4. Taylor expanded in b around inf 78.8%

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

   5. Taylor expanded in b around 0 77.2%

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

   if 5.09999999999999979e-304 < i < 5.5000000000000003e-195

   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. Taylor expanded in x around 0 91.2%

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

   3. Taylor expanded in y around 0 91.2%

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

   4. Taylor expanded in b around inf 82.3%

      \[\leadsto \color{blue}{\log c \cdot b} + \left(a + \left(t + z\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{elif}\;i \leq -3.5 \cdot 10^{-183}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{elif}\;i \leq 5.1 \cdot 10^{-304}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{-195}:\\ \;\;\;\;b \cdot \log c + \left(a + \left(z + t\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 0.000125:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \]

Alternative 10: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + b \cdot \log c\right)\\ t_2 := \mathsf{fma}\left(y, i, z + a\right)\\ t_3 := a + \left(z + x \cdot \log y\right)\\ \mathbf{if}\;i \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -6.5 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-304}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(z + a\right)\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{-88}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 0.00014:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (* b (log c)))))
        (t_2 (fma y i (+ z a)))
        (t_3 (+ a (+ z (* x (log y))))))
   (if (<= i -6.2e+59)
     t_2
     (if (<= i -6.5e-182)
       t_1
       (if (<= i 4.5e-304)
         t_3
         (if (<= i 2.8e-195)
           (+ (* (- b 0.5) (log c)) (+ z a))
           (if (<= i 1.5e-88) t_3 (if (<= i 0.00014) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (z + (b * log(c)));
	double t_2 = fma(y, i, (z + a));
	double t_3 = a + (z + (x * log(y)));
	double tmp;
	if (i <= -6.2e+59) {
		tmp = t_2;
	} else if (i <= -6.5e-182) {
		tmp = t_1;
	} else if (i <= 4.5e-304) {
		tmp = t_3;
	} else if (i <= 2.8e-195) {
		tmp = ((b - 0.5) * log(c)) + (z + a);
	} else if (i <= 1.5e-88) {
		tmp = t_3;
	} else if (i <= 0.00014) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(z + Float64(b * log(c))))
	t_2 = fma(y, i, Float64(z + a))
	t_3 = Float64(a + Float64(z + Float64(x * log(y))))
	tmp = 0.0
	if (i <= -6.2e+59)
		tmp = t_2;
	elseif (i <= -6.5e-182)
		tmp = t_1;
	elseif (i <= 4.5e-304)
		tmp = t_3;
	elseif (i <= 2.8e-195)
		tmp = Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(z + a));
	elseif (i <= 1.5e-88)
		tmp = t_3;
	elseif (i <= 0.00014)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(z + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -6.2e+59], t$95$2, If[LessEqual[i, -6.5e-182], t$95$1, If[LessEqual[i, 4.5e-304], t$95$3, If[LessEqual[i, 2.8e-195], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(z + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.5e-88], t$95$3, If[LessEqual[i, 0.00014], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -6.20000000000000029e59 or 1.3999999999999999e-4 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.5%

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

   5. Taylor expanded in z around inf 70.9%

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

   if -6.20000000000000029e59 < i < -6.49999999999999997e-182 or 1.5e-88 < i < 1.3999999999999999e-4

   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. Taylor expanded in t around 0 79.7%

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

   3. Taylor expanded in y around 0 74.7%

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

   4. Taylor expanded in b around inf 74.7%

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

   5. Taylor expanded in x around 0 64.0%

      \[\leadsto \color{blue}{a + \left(\log c \cdot b + z\right)} \]

   if -6.49999999999999997e-182 < i < 4.4999999999999998e-304 or 2.80000000000000003e-195 < i < 1.5e-88

   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. Taylor expanded in t around 0 82.6%

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

   3. Taylor expanded in y around 0 82.6%

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

   4. Taylor expanded in b around inf 78.8%

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

   5. Taylor expanded in b around 0 77.2%

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

   if 4.4999999999999998e-304 < i < 2.80000000000000003e-195

   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. Taylor expanded in x around 0 91.2%

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

   3. Taylor expanded in y around 0 91.2%

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

   4. Taylor expanded in t around 0 81.4%

      \[\leadsto \left(b - 0.5\right) \cdot \log c + \color{blue}{\left(a + z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{elif}\;i \leq -6.5 \cdot 10^{-182}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-304}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(z + a\right)\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{-88}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 0.00014:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \]

Alternative 11: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + b \cdot \log c\right)\\ t_2 := \mathsf{fma}\left(y, i, z + a\right)\\ t_3 := a + \left(z + x \cdot \log y\right)\\ \mathbf{if}\;i \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -2.5 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-303}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{-195}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{-89}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 0.00014:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (* b (log c)))))
        (t_2 (fma y i (+ z a)))
        (t_3 (+ a (+ z (* x (log y))))))
   (if (<= i -3.9e+59)
     t_2
     (if (<= i -2.5e-182)
       t_1
       (if (<= i 2.6e-303)
         t_3
         (if (<= i 3.7e-195)
           (+ (* (- b 0.5) (log c)) (+ a (+ z t)))
           (if (<= i 1.55e-89) t_3 (if (<= i 0.00014) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (z + (b * log(c)));
	double t_2 = fma(y, i, (z + a));
	double t_3 = a + (z + (x * log(y)));
	double tmp;
	if (i <= -3.9e+59) {
		tmp = t_2;
	} else if (i <= -2.5e-182) {
		tmp = t_1;
	} else if (i <= 2.6e-303) {
		tmp = t_3;
	} else if (i <= 3.7e-195) {
		tmp = ((b - 0.5) * log(c)) + (a + (z + t));
	} else if (i <= 1.55e-89) {
		tmp = t_3;
	} else if (i <= 0.00014) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(z + Float64(b * log(c))))
	t_2 = fma(y, i, Float64(z + a))
	t_3 = Float64(a + Float64(z + Float64(x * log(y))))
	tmp = 0.0
	if (i <= -3.9e+59)
		tmp = t_2;
	elseif (i <= -2.5e-182)
		tmp = t_1;
	elseif (i <= 2.6e-303)
		tmp = t_3;
	elseif (i <= 3.7e-195)
		tmp = Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(a + Float64(z + t)));
	elseif (i <= 1.55e-89)
		tmp = t_3;
	elseif (i <= 0.00014)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(z + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.9e+59], t$95$2, If[LessEqual[i, -2.5e-182], t$95$1, If[LessEqual[i, 2.6e-303], t$95$3, If[LessEqual[i, 3.7e-195], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.55e-89], t$95$3, If[LessEqual[i, 0.00014], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -3.90000000000000021e59 or 1.3999999999999999e-4 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.5%

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

   5. Taylor expanded in z around inf 70.9%

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

   if -3.90000000000000021e59 < i < -2.50000000000000012e-182 or 1.54999999999999998e-89 < i < 1.3999999999999999e-4

   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. Taylor expanded in t around 0 79.7%

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

   3. Taylor expanded in y around 0 74.7%

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

   4. Taylor expanded in b around inf 74.7%

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

   5. Taylor expanded in x around 0 64.0%

      \[\leadsto \color{blue}{a + \left(\log c \cdot b + z\right)} \]

   if -2.50000000000000012e-182 < i < 2.60000000000000005e-303 or 3.69999999999999962e-195 < i < 1.54999999999999998e-89

   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. Taylor expanded in t around 0 82.6%

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

   3. Taylor expanded in y around 0 82.6%

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

   4. Taylor expanded in b around inf 78.8%

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

   5. Taylor expanded in b around 0 77.2%

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

   if 2.60000000000000005e-303 < i < 3.69999999999999962e-195

   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. Taylor expanded in x around 0 91.2%

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

   3. Taylor expanded in y around 0 91.2%

      \[\leadsto \color{blue}{\left(b - 0.5\right) \cdot \log c + \left(a + \left(t + z\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{elif}\;i \leq -2.5 \cdot 10^{-182}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-303}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{-195}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{-89}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;i \leq 0.00014:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \]

Alternative 12: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+253} \lor \neg \left(x \leq 1.85 \cdot 10^{+220}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -6.2e+253) (not (<= x 1.85e+220)))
   (+ a (+ z (* x (log y))))
   (+ (* y i) (+ (* (- b 0.5) (log c)) (+ a (+ z t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -6.2e+253) || !(x <= 1.85e+220)) {
		tmp = a + (z + (x * log(y)));
	} else {
		tmp = (y * i) + (((b - 0.5) * log(c)) + (a + (z + t)));
	}
	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 <= (-6.2d+253)) .or. (.not. (x <= 1.85d+220))) then
        tmp = a + (z + (x * log(y)))
    else
        tmp = (y * i) + (((b - 0.5d0) * log(c)) + (a + (z + t)))
    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 <= -6.2e+253) || !(x <= 1.85e+220)) {
		tmp = a + (z + (x * Math.log(y)));
	} else {
		tmp = (y * i) + (((b - 0.5) * Math.log(c)) + (a + (z + t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -6.2e+253) or not (x <= 1.85e+220):
		tmp = a + (z + (x * math.log(y)))
	else:
		tmp = (y * i) + (((b - 0.5) * math.log(c)) + (a + (z + t)))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -6.2e+253) || !(x <= 1.85e+220))
		tmp = Float64(a + Float64(z + Float64(x * log(y))));
	else
		tmp = Float64(Float64(y * i) + Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(a + Float64(z + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -6.2e+253) || ~((x <= 1.85e+220)))
		tmp = a + (z + (x * log(y)));
	else
		tmp = (y * i) + (((b - 0.5) * log(c)) + (a + (z + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -6.2e+253], N[Not[LessEqual[x, 1.85e+220]], $MachinePrecision]], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000013e253 or 1.85e220 < 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. Taylor expanded in t around 0 99.8%

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

   3. Taylor expanded in y around 0 96.6%

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

   4. Taylor expanded in b around inf 96.6%

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

   5. Taylor expanded in b around 0 93.0%

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

   if -6.20000000000000013e253 < x < 1.85e220

   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. Taylor expanded in x around 0 93.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+253} \lor \neg \left(x \leq 1.85 \cdot 10^{+220}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 13: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+253} \lor \neg \left(x \leq 8 \cdot 10^{+222}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -6.6e+253) (not (<= x 8e+222)))
   (+ a (+ z (* x (log y))))
   (+ (* y i) (+ (* b (log c)) (+ a (+ z t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -6.6e+253) || !(x <= 8e+222)) {
		tmp = a + (z + (x * log(y)));
	} else {
		tmp = (y * i) + ((b * log(c)) + (a + (z + t)));
	}
	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 <= (-6.6d+253)) .or. (.not. (x <= 8d+222))) then
        tmp = a + (z + (x * log(y)))
    else
        tmp = (y * i) + ((b * log(c)) + (a + (z + t)))
    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 <= -6.6e+253) || !(x <= 8e+222)) {
		tmp = a + (z + (x * Math.log(y)));
	} else {
		tmp = (y * i) + ((b * Math.log(c)) + (a + (z + t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -6.6e+253) or not (x <= 8e+222):
		tmp = a + (z + (x * math.log(y)))
	else:
		tmp = (y * i) + ((b * math.log(c)) + (a + (z + t)))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -6.6e+253) || !(x <= 8e+222))
		tmp = Float64(a + Float64(z + Float64(x * log(y))));
	else
		tmp = Float64(Float64(y * i) + Float64(Float64(b * log(c)) + Float64(a + Float64(z + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -6.6e+253) || ~((x <= 8e+222)))
		tmp = a + (z + (x * log(y)));
	else
		tmp = (y * i) + ((b * log(c)) + (a + (z + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -6.6e+253], N[Not[LessEqual[x, 8e+222]], $MachinePrecision]], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5999999999999998e253 or 8.0000000000000004e222 < 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. Taylor expanded in t around 0 99.8%

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

   3. Taylor expanded in y around 0 96.6%

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

   4. Taylor expanded in b around inf 96.6%

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

   5. Taylor expanded in b around 0 93.0%

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

   if -6.5999999999999998e253 < x < 8.0000000000000004e222

   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. Taylor expanded in x around 0 93.5%

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

   3. Taylor expanded in b around inf 92.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+253} \lor \neg \left(x \leq 8 \cdot 10^{+222}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(b \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 14: 58.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.8 \cdot 10^{+59} \lor \neg \left(i \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= i -7.8e+59) (not (<= i 0.00015)))
   (fma y i (+ z a))
   (+ a (+ z (* b (log c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i <= -7.8e+59) || !(i <= 0.00015)) {
		tmp = fma(y, i, (z + a));
	} else {
		tmp = a + (z + (b * log(c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((i <= -7.8e+59) || !(i <= 0.00015))
		tmp = fma(y, i, Float64(z + a));
	else
		tmp = Float64(a + Float64(z + Float64(b * log(c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[i, -7.8e+59], N[Not[LessEqual[i, 0.00015]], $MachinePrecision]], N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision], N[(a + N[(z + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -7.80000000000000043e59 or 1.49999999999999987e-4 < i

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.5%

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

   5. Taylor expanded in z around inf 70.9%

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

   if -7.80000000000000043e59 < i < 1.49999999999999987e-4

   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. Taylor expanded in t around 0 82.1%

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

   3. Taylor expanded in y around 0 79.4%

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

   4. Taylor expanded in b around inf 77.1%

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

   5. Taylor expanded in x around 0 60.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.8 \cdot 10^{+59} \lor \neg \left(i \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + b \cdot \log c\right)\\ \end{array} \]

Alternative 15: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-46} \lor \neg \left(z \leq -3.9 \cdot 10^{-78}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -1.05e+87)
   (+ z (* y i))
   (if (or (<= z -4.4e-46) (not (<= z -3.9e-78)))
     (+ a (* y i))
     (* b (log c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -1.05e+87) {
		tmp = z + (y * i);
	} else if ((z <= -4.4e-46) || !(z <= -3.9e-78)) {
		tmp = a + (y * i);
	} else {
		tmp = b * log(c);
	}
	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 (z <= (-1.05d+87)) then
        tmp = z + (y * i)
    else if ((z <= (-4.4d-46)) .or. (.not. (z <= (-3.9d-78)))) then
        tmp = a + (y * i)
    else
        tmp = b * log(c)
    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 (z <= -1.05e+87) {
		tmp = z + (y * i);
	} else if ((z <= -4.4e-46) || !(z <= -3.9e-78)) {
		tmp = a + (y * i);
	} else {
		tmp = b * Math.log(c);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= -1.05e+87:
		tmp = z + (y * i)
	elif (z <= -4.4e-46) or not (z <= -3.9e-78):
		tmp = a + (y * i)
	else:
		tmp = b * math.log(c)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -1.05e+87)
		tmp = Float64(z + Float64(y * i));
	elseif ((z <= -4.4e-46) || !(z <= -3.9e-78))
		tmp = Float64(a + Float64(y * i));
	else
		tmp = Float64(b * log(c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= -1.05e+87)
		tmp = z + (y * i);
	elseif ((z <= -4.4e-46) || ~((z <= -3.9e-78)))
		tmp = a + (y * i);
	else
		tmp = b * log(c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -1.05e+87], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -4.4e-46], N[Not[LessEqual[z, -3.9e-78]], $MachinePrecision]], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e87

   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. Taylor expanded in z around inf 56.5%

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

   if -1.05e87 < z < -4.4000000000000002e-46 or -3.9000000000000002e-78 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf 42.8%

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

   if -4.4000000000000002e-46 < z < -3.9000000000000002e-78

   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. Taylor expanded in t around 0 66.8%

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

   3. Taylor expanded in y around 0 66.8%

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

   4. Taylor expanded in b around inf 66.8%

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

   5. Taylor expanded in b around inf 66.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-46} \lor \neg \left(z \leq -3.9 \cdot 10^{-78}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]

Alternative 16: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.65 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-46} \lor \neg \left(z \leq -3.9 \cdot 10^{-78}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -3.65e+87)
   (fma y i z)
   (if (or (<= z -4.4e-46) (not (<= z -3.9e-78)))
     (+ a (* y i))
     (* b (log c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -3.65e+87) {
		tmp = fma(y, i, z);
	} else if ((z <= -4.4e-46) || !(z <= -3.9e-78)) {
		tmp = a + (y * i);
	} else {
		tmp = b * log(c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -3.65e+87)
		tmp = fma(y, i, z);
	elseif ((z <= -4.4e-46) || !(z <= -3.9e-78))
		tmp = Float64(a + Float64(y * i));
	else
		tmp = Float64(b * log(c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -3.65e+87], N[(y * i + z), $MachinePrecision], If[Or[LessEqual[z, -4.4e-46], N[Not[LessEqual[z, -3.9e-78]], $MachinePrecision]], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.64999999999999998e87

   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. Taylor expanded in z around inf 56.5%

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

   3. Taylor expanded in z around 0 56.5%

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

      1. +-commutative 56.5%

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

      2. *-commutative 56.5%

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

      3. fma-def 56.5%

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

   5. Simplified 56.5%

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

   if -3.64999999999999998e87 < z < -4.4000000000000002e-46 or -3.9000000000000002e-78 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf 42.8%

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

   if -4.4000000000000002e-46 < z < -3.9000000000000002e-78

   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. Taylor expanded in t around 0 66.8%

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

   3. Taylor expanded in y around 0 66.8%

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

   4. Taylor expanded in b around inf 66.8%

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

   5. Taylor expanded in b around inf 66.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.65 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-46} \lor \neg \left(z \leq -3.9 \cdot 10^{-78}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]

Alternative 17: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+205} \lor \neg \left(b \leq 1.95 \cdot 10^{+239}\right):\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= b -1.9e+205) (not (<= b 1.95e+239)))
   (* b (log c))
   (fma y i (+ z a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((b <= -1.9e+205) || !(b <= 1.95e+239)) {
		tmp = b * log(c);
	} else {
		tmp = fma(y, i, (z + a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((b <= -1.9e+205) || !(b <= 1.95e+239))
		tmp = Float64(b * log(c));
	else
		tmp = fma(y, i, Float64(z + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[b, -1.9e+205], N[Not[LessEqual[b, 1.95e+239]], $MachinePrecision]], N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision], N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9e205 or 1.9499999999999999e239 < b

   1. Initial program 99.7%

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

   2. Taylor expanded in t around 0 99.7%

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

   3. Taylor expanded in y around 0 89.5%

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

   4. Taylor expanded in b around inf 89.5%

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

   5. Taylor expanded in b around inf 71.8%

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

   if -1.9e205 < b < 1.9499999999999999e239

   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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. associate-+l+ 99.9%

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

      11. fma-def 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 84.7%

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

   5. Taylor expanded in z around inf 58.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+205} \lor \neg \left(b \leq 1.95 \cdot 10^{+239}\right):\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z + a\right)\\ \end{array} \]

Alternative 18: 39.9% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+246}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+197} \lor \neg \left(z \leq -4.9 \cdot 10^{+132}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -1.4e+246)
   z
   (if (or (<= z -3.1e+197) (not (<= z -4.9e+132))) (+ a (* y i)) z)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -1.4e+246) {
		tmp = z;
	} else if ((z <= -3.1e+197) || !(z <= -4.9e+132)) {
		tmp = a + (y * i);
	} else {
		tmp = z;
	}
	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 (z <= (-1.4d+246)) then
        tmp = z
    else if ((z <= (-3.1d+197)) .or. (.not. (z <= (-4.9d+132)))) then
        tmp = a + (y * i)
    else
        tmp = z
    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 (z <= -1.4e+246) {
		tmp = z;
	} else if ((z <= -3.1e+197) || !(z <= -4.9e+132)) {
		tmp = a + (y * i);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= -1.4e+246:
		tmp = z
	elif (z <= -3.1e+197) or not (z <= -4.9e+132):
		tmp = a + (y * i)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -1.4e+246)
		tmp = z;
	elseif ((z <= -3.1e+197) || !(z <= -4.9e+132))
		tmp = Float64(a + Float64(y * i));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= -1.4e+246)
		tmp = z;
	elseif ((z <= -3.1e+197) || ~((z <= -4.9e+132)))
		tmp = a + (y * i);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -1.4e+246], z, If[Or[LessEqual[z, -3.1e+197], N[Not[LessEqual[z, -4.9e+132]], $MachinePrecision]], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e246 or -3.1e197 < z < -4.9000000000000002e132

   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. Taylor expanded in x around 0 87.8%

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

   3. Taylor expanded in y around 0 80.4%

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

   4. Taylor expanded in z around inf 53.0%

      \[\leadsto \color{blue}{z} \]

   if -1.39999999999999994e246 < z < -3.1e197 or -4.9000000000000002e132 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf 41.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+246}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+197} \lor \neg \left(z \leq -4.9 \cdot 10^{+132}\right):\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 19: 42.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -3.6e+87) (+ z (* y i)) (+ a (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -3.6e+87) {
		tmp = z + (y * i);
	} else {
		tmp = a + (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 (z <= (-3.6d+87)) then
        tmp = z + (y * i)
    else
        tmp = a + (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 (z <= -3.6e+87) {
		tmp = z + (y * i);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= -3.6e+87:
		tmp = z + (y * i)
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -3.6e+87)
		tmp = Float64(z + Float64(y * i));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= -3.6e+87)
		tmp = z + (y * i);
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -3.6e+87], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999994e87

   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. Taylor expanded in z around inf 56.5%

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

   if -3.59999999999999994e87 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf 42.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+87}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 20: 20.8% accurate, 71.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+87}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (if (<= z -2.95e+87) z a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2.95e+87) {
		tmp = z;
	} else {
		tmp = a;
	}
	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 (z <= (-2.95d+87)) then
        tmp = z
    else
        tmp = a
    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 (z <= -2.95e+87) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= -2.95e+87:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2.95e+87)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= -2.95e+87)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2.95e+87], z, a]

Derivation

1. Split input into 2 regimes

2. if z < -2.9499999999999998e87

   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. Taylor expanded in x around 0 86.6%

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

   3. Taylor expanded in y around 0 73.2%

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

   4. Taylor expanded in z around inf 42.7%

      \[\leadsto \color{blue}{z} \]

   if -2.9499999999999998e87 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf 42.2%

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

   3. Taylor expanded in a around inf 19.4%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+87}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 21: 16.2% accurate, 219.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in a around inf 39.4%

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

3. Taylor expanded in a around inf 18.1%

   \[\leadsto \color{blue}{a} \]

4. Final simplification 18.1%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023274 
(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)))