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

Percentage Accurate: 99.8% → 99.8%

Time: 16.2s

Alternatives: 25

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

   11. sub-neg N/A

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

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

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

   13. metadata-eval N/A

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

   14. log-lowering-log.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ t_2 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+251}:\\ \;\;\;\;y \cdot i + \left(z + t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) (- b 0.5)))
        (t_2 (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) t_1))))
   (if (<= t_2 -4e+251)
     (+ (* y i) (+ z t_1))
     (if (<= t_2 -2e+21)
       (+ (* y i) (fma (log y) x z))
       (+ (* y i) (fma (log y) x a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * (b - 0.5);
	double t_2 = (y * i) + ((a + (t + (z + (log(y) * x)))) + t_1);
	double tmp;
	if (t_2 <= -4e+251) {
		tmp = (y * i) + (z + t_1);
	} else if (t_2 <= -2e+21) {
		tmp = (y * i) + fma(log(y), x, z);
	} else {
		tmp = (y * i) + fma(log(y), x, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * Float64(b - 0.5))
	t_2 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + t_1))
	tmp = 0.0
	if (t_2 <= -4e+251)
		tmp = Float64(Float64(y * i) + Float64(z + t_1));
	elseif (t_2 <= -2e+21)
		tmp = Float64(Float64(y * i) + fma(log(y), x, z));
	else
		tmp = Float64(Float64(y * i) + fma(log(y), x, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+251], N[(N[(y * i), $MachinePrecision] + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+21], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + a), $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)) < -4.0000000000000002e251

   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 61.0%

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

   if -4.0000000000000002e251 < (+.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)) < -2e21

   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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 55.2%

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

   if -2e21 < (+.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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 54.1%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -4 \cdot 10^{+251}:\\ \;\;\;\;y \cdot i + \left(z + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(z + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))))
   (if (<= t_1 (- INFINITY))
     (* y i)
     (if (<= t_1 2e+93) (+ (+ z t) a) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 2e+93) {
		tmp = (z + t) + a;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 2e+93)
		tmp = Float64(Float64(z + t) + a);
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 2e+93], N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 93.8

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

   5. Simplified 93.8%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.00000000000000009e93

   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 82.4

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

   5. Simplified 82.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 69.0%

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

   9. Taylor expanded in i around 0

      \[\leadsto a + \left(\color{blue}{z} + t\right) \]
   10. Step-by-step derivation

   11. Simplified 60.3%

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

   if 2.00000000000000009e93 < (+.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 86.2

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

   5. Simplified 86.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 86.2

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

   7. Applied egg-rr 86.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   9. Step-by-step derivation

   10. Simplified 42.9%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(z + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(z + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))))
   (if (<= t_1 (- INFINITY))
     (* y i)
     (if (<= t_1 2e+307) (+ (+ z 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 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 2e+307) {
		tmp = (z + t) + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((a + (t + (z + (Math.log(y) * x)))) + (Math.log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * i;
	} else if (t_1 <= 2e+307) {
		tmp = (z + t) + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * i) + ((a + (t + (z + (math.log(y) * x)))) + (math.log(c) * (b - 0.5)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * i
	elif t_1 <= 2e+307:
		tmp = (z + t) + a
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 2e+307)
		tmp = Float64(Float64(z + t) + 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 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * i;
	elseif (t_1 <= 2e+307)
		tmp = (z + t) + 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[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(N[(z + t), $MachinePrecision] + 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)) < -inf.0 or 1.99999999999999997e307 < (+.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 94.3

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

   5. Simplified 94.3%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.99999999999999997e307

   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 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 67.8%

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

   9. Taylor expanded in i around 0

      \[\leadsto a + \left(\color{blue}{z} + t\right) \]
   10. Step-by-step derivation

   11. Simplified 57.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(z + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (* y i)
          (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))))
   (if (<= t_1 (- INFINITY)) (* y i) (if (<= t_1 2e+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 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 2e+307) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + ((a + (t + (z + (Math.log(y) * x)))) + (Math.log(c) * (b - 0.5)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * i;
	} else if (t_1 <= 2e+307) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * i) + ((a + (t + (z + (math.log(y) * x)))) + (math.log(c) * (b - 0.5)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * i
	elif t_1 <= 2e+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(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 2e+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 = (y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * i;
	elseif (t_1 <= 2e+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[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 2e+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)) < -inf.0 or 1.99999999999999997e307 < (+.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 94.3

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

   5. Simplified 94.3%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.99999999999999997e307

   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 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 37.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -2e+21)
   (+ (* y i) (fma (log y) x z))
   (+ (* y i) (fma (log y) x a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -2e+21) {
		tmp = (y * i) + fma(log(y), x, z);
	} else {
		tmp = (y * i) + fma(log(y), x, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -2e+21)
		tmp = Float64(Float64(y * i) + fma(log(y), x, z));
	else
		tmp = Float64(Float64(y * i) + fma(log(y), x, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+21], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + a), $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)) < -2e21

   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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 55.8%

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

   if -2e21 < (+.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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 54.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -5e+19)
   (fma y i z)
   (+ a (fma i y t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19) {
		tmp = fma(y, i, z);
	} else {
		tmp = a + fma(i, y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -5e+19)
		tmp = fma(y, i, z);
	else
		tmp = Float64(a + fma(i, y, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+19], N[(y * i + z), $MachinePrecision], N[(a + N[(i * y + t), $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)) < -5e19

   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 85.0

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

   5. Simplified 85.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 85.0

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 41.6%

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

   if -5e19 < (+.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 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 70.4%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 58.9

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

   11. Simplified 58.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -5e+19)
   (fma y i z)
   (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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19) {
		tmp = fma(y, i, z);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -5e+19)
		tmp = fma(y, i, z);
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+19], N[(y * i + z), $MachinePrecision], N[(y * i + 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)) < -5e19

   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 85.0

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

   5. Simplified 85.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 85.0

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 41.6%

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

   if -5e19 < (+.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 85.6

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

   5. Simplified 85.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 85.6

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

   7. Applied egg-rr 85.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   9. Step-by-step derivation

   10. Simplified 40.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -5e+19)
   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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19) {
		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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5d0)))) <= (-5d+19)) 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 (((y * i) + ((a + (t + (z + (Math.log(y) * x)))) + (Math.log(c) * (b - 0.5)))) <= -5e+19) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((y * i) + ((a + (t + (z + (math.log(y) * x)))) + (math.log(c) * (b - 0.5)))) <= -5e+19:
		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(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -5e+19)
		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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19)
		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[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+19], 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)) < -5e19

   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 22.5%

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

   if -5e19 < (+.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 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 35.2%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 16.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (* y i) (+ (+ a (+ t (+ z (* (log y) x)))) (* (log c) (- b 0.5))))
      -5e+19)
   z
   a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19) {
		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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5d0)))) <= (-5d+19)) 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 (((y * i) + ((a + (t + (z + (Math.log(y) * x)))) + (Math.log(c) * (b - 0.5)))) <= -5e+19) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((y * i) + ((a + (t + (z + (math.log(y) * x)))) + (math.log(c) * (b - 0.5)))) <= -5e+19:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(y * i) + Float64(Float64(a + Float64(t + Float64(z + Float64(log(y) * x)))) + Float64(log(c) * Float64(b - 0.5)))) <= -5e+19)
		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 (((y * i) + ((a + (t + (z + (log(y) * x)))) + (log(c) * (b - 0.5)))) <= -5e+19)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(y * i), $MachinePrecision] + N[(N[(a + N[(t + N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+19], 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)) < -5e19

   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 22.5%

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

   if -5e19 < (+.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 16.8%

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

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot i + \left(\left(a + \left(t + \left(z + \log y \cdot x\right)\right)\right) + \log c \cdot \left(b - 0.5\right)\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 2.4e-51)
   (+ a (+ (fma (log c) (+ b -0.5) z) (fma x (log y) t)))
   (fma y i (+ (+ z t) (fma (+ b -0.5) (log c) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 2.4e-51) {
		tmp = a + (fma(log(c), (b + -0.5), z) + fma(x, log(y), t));
	} else {
		tmp = fma(y, i, ((z + t) + fma((b + -0.5), log(c), a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 2.4e-51)
		tmp = Float64(a + Float64(fma(log(c), Float64(b + -0.5), z) + fma(x, log(y), t)));
	else
		tmp = fma(y, i, Float64(Float64(z + t) + fma(Float64(b + -0.5), log(c), a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 2.4e-51], N[(a + N[(N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(z + t), $MachinePrecision] + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e-51

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 99.8%

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

   if 2.4e-51 < y

   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 88.7

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

   5. Simplified 88.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 88.7

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

   7. Applied egg-rr 88.7%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. sub-neg N/A

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

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

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

   7. metadata-eval N/A

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

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

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

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

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

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

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

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

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

   15. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 13: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in t around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. +-commutative N/A

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

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

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

   15. log-lowering-log.f64 82.0

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

5. Simplified 82.0%

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

Alternative 14: 90.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (fma (log y) x z))))
   (if (<= x -2.9e+172)
     t_1
     (if (<= x 1.35e+239)
       (fma y i (+ (+ z t) (fma (+ b -0.5) (log c) a)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * i) + fma(log(y), x, z);
	double tmp;
	if (x <= -2.9e+172) {
		tmp = t_1;
	} else if (x <= 1.35e+239) {
		tmp = fma(y, i, ((z + t) + fma((b + -0.5), log(c), a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * i) + fma(log(y), x, z))
	tmp = 0.0
	if (x <= -2.9e+172)
		tmp = t_1;
	elseif (x <= 1.35e+239)
		tmp = fma(y, i, Float64(Float64(z + t) + fma(Float64(b + -0.5), log(c), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+172], t$95$1, If[LessEqual[x, 1.35e+239], N[(y * i + N[(N[(z + t), $MachinePrecision] + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999999e172 or 1.3499999999999999e239 < 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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 75.3%

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

   if -2.8999999999999999e172 < x < 1.3499999999999999e239

   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 94.6

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

   5. Simplified 94.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      14. log-lowering-log.f64 94.6

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+172}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + \mathsf{fma}\left(b + -0.5, \log c, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+239}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y i) (fma (log y) x z))))
   (if (<= x -8.2e+172)
     t_1
     (if (<= x 1.35e+239)
       (+ 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 = (y * i) + fma(log(y), x, z);
	double tmp;
	if (x <= -8.2e+172) {
		tmp = t_1;
	} else if (x <= 1.35e+239) {
		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(Float64(y * i) + fma(log(y), x, z))
	tmp = 0.0
	if (x <= -8.2e+172)
		tmp = t_1;
	elseif (x <= 1.35e+239)
		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[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e+172], t$95$1, If[LessEqual[x, 1.35e+239], 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 < -8.200000000000001e172 or 1.3499999999999999e239 < 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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 75.3%

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

   if -8.200000000000001e172 < x < 1.3499999999999999e239

   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 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+172}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+239}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b, y \cdot i\right)\\ \mathbf{if}\;b \leq -3.75 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) b (* y i))))
   (if (<= b -3.75e+202)
     t_1
     (if (<= b 7.5e+145) (fma y i (+ (+ z t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), b, (y * i));
	double tmp;
	if (b <= -3.75e+202) {
		tmp = t_1;
	} else if (b <= 7.5e+145) {
		tmp = fma(y, i, ((z + t) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), b, Float64(y * i))
	tmp = 0.0
	if (b <= -3.75e+202)
		tmp = t_1;
	elseif (b <= 7.5e+145)
		tmp = fma(y, i, Float64(Float64(z + t) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.75e+202], t$95$1, If[LessEqual[b, 7.5e+145], N[(y * i + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.75e202 or 7.50000000000000006e145 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 73.9

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

   5. Simplified 73.9%

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

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

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

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

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

      3. *-lowering-*.f64 73.9

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

   7. Applied egg-rr 73.9%

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

   if -3.75e202 < b < 7.50000000000000006e145

   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 83.7

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

   5. Simplified 83.7%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 82.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

      7. +-lowering-+.f64 82.2

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

   10. Applied egg-rr 82.2%

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

Alternative 17: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -9.5e+95) (+ a (+ t (fma i y z))) (+ (* y i) (fma (log y) x a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -9.5e+95) {
		tmp = a + (t + fma(i, y, z));
	} else {
		tmp = (y * i) + fma(log(y), x, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -9.5e+95)
		tmp = Float64(a + Float64(t + fma(i, y, z)));
	else
		tmp = Float64(Float64(y * i) + fma(log(y), x, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -9.5e+95], N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000004e95

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 76.9%

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

   if -9.5000000000000004e95 < z

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 56.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \mathsf{fma}\left(\log y, x, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 71.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))))
   (if (<= b -2.2e+249)
     t_1
     (if (<= b 1.95e+158) (fma y i (+ (+ z t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double tmp;
	if (b <= -2.2e+249) {
		tmp = t_1;
	} else if (b <= 1.95e+158) {
		tmp = fma(y, i, ((z + t) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	tmp = 0.0
	if (b <= -2.2e+249)
		tmp = t_1;
	elseif (b <= 1.95e+158)
		tmp = fma(y, i, Float64(Float64(z + t) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.2e+249], t$95$1, If[LessEqual[b, 1.95e+158], N[(y * i + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.1999999999999998e249 or 1.95e158 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 63.7

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

   5. Simplified 63.7%

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

   if -2.1999999999999998e249 < b < 1.95e158

   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 83.8

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

   5. Simplified 83.8%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 80.1%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

      7. +-lowering-+.f64 80.1

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

   10. Applied egg-rr 80.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+249}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+240}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -5.6e+240) t_1 (if (<= x 6.5e+250) (fma y i (+ (+ z t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -5.6e+240) {
		tmp = t_1;
	} else if (x <= 6.5e+250) {
		tmp = fma(y, i, ((z + t) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -5.6e+240)
		tmp = t_1;
	elseif (x <= 6.5e+250)
		tmp = fma(y, i, Float64(Float64(z + t) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.6e+240], t$95$1, If[LessEqual[x, 6.5e+250], N[(y * i + N[(N[(z + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000002e240 or 6.5000000000000004e250 < 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 x around inf

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

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

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

      2. log-lowering-log.f64 71.6

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

   5. Simplified 71.6%

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

   if -5.6000000000000002e240 < x < 6.5000000000000004e250

   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 92.8

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

   5. Simplified 92.8%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 77.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

      7. +-lowering-+.f64 77.7

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

   10. Applied egg-rr 77.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+240}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(z + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 67.4% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-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 85.3

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

5. Simplified 85.3%

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

6. Taylor expanded in t around inf

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

8. Simplified 71.4%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

   7. +-lowering-+.f64 71.4

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

10. Applied egg-rr 71.4%

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

Alternative 21: 67.4% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. 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 85.3

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

5. Simplified 85.3%

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

6. Taylor expanded in t around inf

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

8. Simplified 71.4%

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

9. Final simplification 71.4%

   \[\leadsto a + \left(t + \mathsf{fma}\left(i, y, z\right)\right) \]
10. Add Preprocessing

Alternative 22: 52.6% accurate, 23.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-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 85.3

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

5. Simplified 85.3%

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

6. Taylor expanded in t around inf

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

8. Simplified 71.4%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

   7. +-lowering-+.f64 71.4

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

10. Applied egg-rr 71.4%

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

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. +-lowering-+.f64 53.7

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

13. Simplified 53.7%

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

Alternative 23: 52.6% accurate, 23.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-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 85.3

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

5. Simplified 85.3%

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

6. Taylor expanded in t around inf

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

8. Simplified 71.4%

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

9. Taylor expanded in t around 0

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

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

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

    2. +-commutative N/A

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

    3. accelerator-lowering-fma.f64 53.7

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

11. Simplified 53.7%

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

Alternative 24: 31.2% 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 85.3

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

5. Simplified 85.3%

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

6. Taylor expanded in z around inf

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

8. Simplified 32.5%

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

9. Final simplification 32.5%

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

Alternative 25: 16.3% 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 16.5%

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

Reproduce

herbie shell --seed 2024198 
(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)))