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

Percentage Accurate: 99.8% → 99.8%

Time: 15.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 65.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ t_2 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\log c, -0.5, \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma x (log y) z) (+ t a)))
        (t_2
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_2 -7e+307)
     (* y i)
     (if (<= t_2 -1e+109)
       t_1
       (if (<= t_2 5e+44)
         (fma (log c) -0.5 (fma i y z))
         (if (<= t_2 1e+262) t_1 (+ a (* y i))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, log(y), z) + (t + a);
	double t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -7e+307) {
		tmp = y * i;
	} else if (t_2 <= -1e+109) {
		tmp = t_1;
	} else if (t_2 <= 5e+44) {
		tmp = fma(log(c), -0.5, fma(i, y, z));
	} else if (t_2 <= 1e+262) {
		tmp = t_1;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(x, log(y), z) + Float64(t + a))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -7e+307)
		tmp = Float64(y * i);
	elseif (t_2 <= -1e+109)
		tmp = t_1;
	elseif (t_2 <= 5e+44)
		tmp = fma(log(c), -0.5, fma(i, y, z));
	elseif (t_2 <= 1e+262)
		tmp = t_1;
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -7e+307], N[(y * i), $MachinePrecision], If[LessEqual[t$95$2, -1e+109], t$95$1, If[LessEqual[t$95$2, 5e+44], N[(N[Log[c], $MachinePrecision] * -0.5 + N[(i * y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+262], t$95$1, N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -7.00000000000000028e307

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -7.00000000000000028e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -9.99999999999999982e108 or 4.9999999999999996e44 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1e262

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 71.5%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 89.9%

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

   11. Taylor expanded in y around 0

       \[\leadsto \color{blue}{a + \left(t + \left(z + x \cdot \log y\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. +-commutative N/A

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

       5. +-lowering-+.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. +-lowering-+.f64 83.1

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

   13. Simplified 83.1%

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

   if -9.99999999999999982e108 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.9999999999999996e44

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 75.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 64.4

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

   8. Simplified 64.4%

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

   if 1e262 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 59.6%

      \[\leadsto \color{blue}{a} + y \cdot i \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\log c, -0.5, \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -7e+307)
     (* y i)
     (if (<= t_1 1e+262) (+ (fma x (log y) z) (+ t a)) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -7e+307) {
		tmp = y * i;
	} else if (t_1 <= 1e+262) {
		tmp = fma(x, log(y), z) + (t + a);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -7e+307)
		tmp = Float64(y * i);
	elseif (t_1 <= 1e+262)
		tmp = Float64(fma(x, log(y), z) + Float64(t + a));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -7e+307], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 1e+262], N[(N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -7.00000000000000028e307

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -7.00000000000000028e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1e262

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 72.2%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 86.9%

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

   11. Taylor expanded in y around 0

       \[\leadsto \color{blue}{a + \left(t + \left(z + x \cdot \log y\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. +-commutative N/A

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

       5. +-lowering-+.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. +-lowering-+.f64 78.1

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

   13. Simplified 78.1%

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

   if 1e262 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 59.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -1e+233)
     (fma y i z)
     (if (<= t_1 50.0) (fma (log c) (+ b -0.5) z) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -1e+233) {
		tmp = fma(y, i, z);
	} else if (t_1 <= 50.0) {
		tmp = fma(log(c), (b + -0.5), z);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -1e+233)
		tmp = fma(y, i, z);
	elseif (t_1 <= 50.0)
		tmp = fma(log(c), Float64(b + -0.5), z);
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+233], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, 50.0], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -9.99999999999999974e232

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 41.4%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 41.4

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

   7. Applied egg-rr 41.4%

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

   if -9.99999999999999974e232 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 49.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 39.8

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

   8. Simplified 39.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 49.2%

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

Alternative 5: 40.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 5e+44) (fma y i z) (if (<= t_1 5e+307) (+ t a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= 5e+44) {
		tmp = fma(y, i, z);
	} else if (t_1 <= 5e+307) {
		tmp = t + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= 5e+44)
		tmp = fma(y, i, z);
	elseif (t_1 <= 5e+307)
		tmp = Float64(t + a);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+44], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], N[(t + a), $MachinePrecision], N[(y * i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.9999999999999996e44

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 36.5%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 36.5

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

   7. Applied egg-rr 36.5%

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

   if 4.9999999999999996e44 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5e307

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.9

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

   5. Simplified 83.9%

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

   6. Taylor expanded in t around inf

      \[\leadsto a + \color{blue}{t} \]
   7. Step-by-step derivation

   8. Simplified 45.1%

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

   if 5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 95.1

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

   5. Simplified 95.1%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -7e+307) (* y i) (if (<= t_1 5e+307) (+ z a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -7e+307) {
		tmp = y * i;
	} else if (t_1 <= 5e+307) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-7d+307)) then
        tmp = y * i
    else if (t_1 <= 5d+307) then
        tmp = z + a
    else
        tmp = y * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -7e+307) {
		tmp = y * i;
	} else if (t_1 <= 5e+307) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -7e+307:
		tmp = y * i
	elif t_1 <= 5e+307:
		tmp = z + a
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -7e+307)
		tmp = Float64(y * i);
	elseif (t_1 <= 5e+307)
		tmp = Float64(z + a);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -7e+307)
		tmp = y * i;
	elseif (t_1 <= 5e+307)
		tmp = z + a;
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -7e+307], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], N[(z + a), $MachinePrecision], N[(y * i), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -7.00000000000000028e307 or 5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 97.1

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

   5. Simplified 97.1%

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

   if -7.00000000000000028e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5e307

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 82.6

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

   5. Simplified 82.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto a + \color{blue}{z} \]
   7. Step-by-step derivation

   8. Simplified 39.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -7 \cdot 10^{+307}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+307}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      50.0)
   (fma y i z)
   (+ 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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 50.0) {
		tmp = fma(y, i, z);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 50.0], N[(y * i + z), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 36.1%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 36.1

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

   7. Applied egg-rr 36.1%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 49.2%

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

Alternative 8: 23.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -10.0) {
		tmp = z;
	} else {
		tmp = t + 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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-10.0d0)) then
        tmp = z
    else
        tmp = t + 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 (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -10.0) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 20.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto a + \color{blue}{t} \]
   7. Step-by-step derivation

   8. Simplified 36.7%

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

4. Final simplification 27.6%

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

Alternative 9: 16.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -10.0) {
		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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-10.0d0)) 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 (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -10.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -10.0)
		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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -10.0)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 20.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Simplified 22.0%

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

Alternative 10: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + t\_1\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= t_1 5e+194)
     (+ t (+ z (fma x (log y) (fma y i a))))
     (+ (+ z t_1) (* y i)))))

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 (t_1 <= 5e+194) {
		tmp = t + (z + fma(x, log(y), fma(y, i, a)));
	} else {
		tmp = (z + t_1) + (y * i);
	}
	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 (t_1 <= 5e+194)
		tmp = Float64(t + Float64(z + fma(x, log(y), fma(y, i, a))));
	else
		tmp = Float64(Float64(z + t_1) + Float64(y * i));
	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[t$95$1, 5e+194], N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(y * i + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 4.99999999999999989e194

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 75.7%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 92.5%

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

       1. *-inverses N/A

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

       2. div-inv N/A

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

       3. remove-double-div N/A

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

       4. +-lowering-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. +-lowering-+.f64 N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. log-lowering-log.f64 N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 92.6

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

   12. Applied egg-rr 92.6%

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

   if 4.99999999999999989e194 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 77.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq 5 \cdot 10^{+194}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 5.6e+45)
   (+ (fma i y z) (fma (log c) (+ b -0.5) (fma x (log y) t)))
   (+ t (+ z (fma x (log y) (fma y i a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 5.6e+45) {
		tmp = fma(i, y, z) + fma(log(c), (b + -0.5), fma(x, log(y), t));
	} else {
		tmp = t + (z + fma(x, log(y), fma(y, i, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 5.6e+45)
		tmp = Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), fma(x, log(y), t)));
	else
		tmp = Float64(t + Float64(z + fma(x, log(y), fma(y, i, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 5.6e+45], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(y * i + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.5999999999999999e45

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. log-lowering-log.f64 88.3

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

   5. Simplified 88.3%

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

   if 5.5999999999999999e45 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 75.7%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 94.4%

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

       1. *-inverses N/A

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

       2. div-inv N/A

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

       3. remove-double-div N/A

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

       4. +-lowering-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. +-lowering-+.f64 N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. log-lowering-log.f64 N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 94.6

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

   12. Applied egg-rr 94.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+129}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ t (+ z (fma x (log y) (fma y i a))))))
   (if (<= x -8.6e+80)
     t_1
     (if (<= x 8.5e+129)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t + (z + fma(x, log(y), fma(y, i, a)));
	double tmp;
	if (x <= -8.6e+80) {
		tmp = t_1;
	} else if (x <= 8.5e+129) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t + Float64(z + fma(x, log(y), fma(y, i, a))))
	tmp = 0.0
	if (x <= -8.6e+80)
		tmp = t_1;
	elseif (x <= 8.5e+129)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(y * i + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.6e+80], t$95$1, If[LessEqual[x, 8.5e+129], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.60000000000000008e80 or 8.5000000000000001e129 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 70.4%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 94.1%

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

       1. *-inverses N/A

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

       2. div-inv N/A

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

       3. remove-double-div N/A

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

       4. +-lowering-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. +-lowering-+.f64 N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. log-lowering-log.f64 N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 94.2

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

   12. Applied egg-rr 94.2%

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

   if -8.60000000000000008e80 < x < 8.5000000000000001e129

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+80}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+129}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (- b 0.5) -1e+223)
   (fma (log c) (+ b -0.5) z)
   (+ t (+ z (fma x (log y) (fma y i a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((b - 0.5) <= -1e+223) {
		tmp = fma(log(c), (b + -0.5), z);
	} else {
		tmp = t + (z + fma(x, log(y), fma(y, i, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(b - 0.5) <= -1e+223)
		tmp = fma(log(c), Float64(b + -0.5), z);
	else
		tmp = Float64(t + Float64(z + fma(x, log(y), fma(y, i, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(b - 0.5), $MachinePrecision], -1e+223], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision], N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(y * i + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -1.00000000000000005e223

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 91.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 75.9

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

   8. Simplified 75.9%

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

   if -1.00000000000000005e223 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

   5. Simplified 75.5%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 91.8%

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

       1. *-inverses N/A

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

       2. div-inv N/A

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

       3. remove-double-div N/A

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

       4. +-lowering-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. +-lowering-+.f64 N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. log-lowering-log.f64 N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 92.0

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

   12. Applied egg-rr 92.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(y, i, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.7% accurate, 58.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+l+ N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. log-lowering-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-lowering-+.f64 84.5

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

5. Simplified 84.5%

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

6. Taylor expanded in z around inf

   \[\leadsto a + \color{blue}{z} \]
7. Step-by-step derivation

8. Simplified 35.2%

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

9. Final simplification 35.2%

   \[\leadsto z + a \]
10. Add Preprocessing

Alternative 15: 16.5% accurate, 234.0× 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. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
4. Step-by-step derivation

5. Simplified 19.6%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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)))