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

Percentage Accurate: 99.8% → 99.8%

Time: 13.4s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(b - 0.5\right) \cdot \log c + \left(\left(\left(t\_1 + z\right) + t\right) + a\right)\right) + i \cdot y\\ t_3 := \mathsf{fma}\left(y, i, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (* (- b 0.5) (log c)) (+ (+ (+ t_1 z) t) a)) (* i y)))
        (t_3 (fma y i t_1)))
   (if (<= t_2 -5e+304)
     t_3
     (if (<= t_2 1e+306) (+ (fma (- b 0.5) (log c) z) a) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = (((b - 0.5) * log(c)) + (((t_1 + z) + t) + a)) + (i * y);
	double t_3 = fma(y, i, t_1);
	double tmp;
	if (t_2 <= -5e+304) {
		tmp = t_3;
	} else if (t_2 <= 1e+306) {
		tmp = fma((b - 0.5), log(c), z) + a;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(b - 0.5) * log(c)) + Float64(Float64(Float64(t_1 + z) + t) + a)) + Float64(i * y))
	t_3 = fma(y, i, t_1)
	tmp = 0.0
	if (t_2 <= -5e+304)
		tmp = t_3;
	elseif (t_2 <= 1e+306)
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + a);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * i + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+304], t$95$3, If[LessEqual[t$95$2, 1e+306], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], t$95$3]]]]]

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)) < -4.9999999999999997e304 or 1.00000000000000002e306 < (+.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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

   if -4.9999999999999997e304 < (+.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)) < 1.00000000000000002e306

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 56.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b - 0.5\right) \cdot \log c + \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)\right) + i \cdot y \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{elif}\;\left(\left(b - 0.5\right) \cdot \log c + \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)\right) + i \cdot y \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c + \left(a + t\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+81}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* b (log c)) (+ a t))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+91) t_1 (if (<= t_2 1e+81) (+ (* i y) (+ a t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b * log(c)) + (a + t);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+91) {
		tmp = t_1;
	} else if (t_2 <= 1e+81) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (b * log(c)) + (a + t)
    t_2 = (b - 0.5d0) * log(c)
    if (t_2 <= (-2d+91)) then
        tmp = t_1
    else if (t_2 <= 1d+81) then
        tmp = (i * y) + (a + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b * Math.log(c)) + (a + t);
	double t_2 = (b - 0.5) * Math.log(c);
	double tmp;
	if (t_2 <= -2e+91) {
		tmp = t_1;
	} else if (t_2 <= 1e+81) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (b * math.log(c)) + (a + t)
	t_2 = (b - 0.5) * math.log(c)
	tmp = 0
	if t_2 <= -2e+91:
		tmp = t_1
	elif t_2 <= 1e+81:
		tmp = (i * y) + (a + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b * log(c)) + Float64(a + t))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+91)
		tmp = t_1;
	elseif (t_2 <= 1e+81)
		tmp = Float64(Float64(i * y) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (b * log(c)) + (a + t);
	t_2 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (t_2 <= -2e+91)
		tmp = t_1;
	elseif (t_2 <= 1e+81)
		tmp = (i * y) + (a + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+91], t$95$1, If[LessEqual[t$95$2, 1e+81], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2.00000000000000016e91 or 9.99999999999999921e80 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 68.1%

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

   if -2.00000000000000016e91 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.99999999999999921e80

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 71.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+91}:\\ \;\;\;\;b \cdot \log c + \left(a + t\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+81}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c + a\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* b (log c)) a)) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+91) t_1 (if (<= t_2 5e+138) (+ (* i y) (+ a t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b * log(c)) + a;
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+91) {
		tmp = t_1;
	} else if (t_2 <= 5e+138) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (b * log(c)) + a
    t_2 = (b - 0.5d0) * log(c)
    if (t_2 <= (-2d+91)) then
        tmp = t_1
    else if (t_2 <= 5d+138) then
        tmp = (i * y) + (a + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b * Math.log(c)) + a;
	double t_2 = (b - 0.5) * Math.log(c);
	double tmp;
	if (t_2 <= -2e+91) {
		tmp = t_1;
	} else if (t_2 <= 5e+138) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (b * math.log(c)) + a
	t_2 = (b - 0.5) * math.log(c)
	tmp = 0
	if t_2 <= -2e+91:
		tmp = t_1
	elif t_2 <= 5e+138:
		tmp = (i * y) + (a + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b * log(c)) + a)
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+91)
		tmp = t_1;
	elseif (t_2 <= 5e+138)
		tmp = Float64(Float64(i * y) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (b * log(c)) + a;
	t_2 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (t_2 <= -2e+91)
		tmp = t_1;
	elseif (t_2 <= 5e+138)
		tmp = (i * y) + (a + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+91], t$95$1, If[LessEqual[t$95$2, 5e+138], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2.00000000000000016e91 or 5.00000000000000016e138 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto b \cdot \log c + a \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto b \cdot \log c + a \]

   if -2.00000000000000016e91 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 5.00000000000000016e138

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+91}:\\ \;\;\;\;b \cdot \log c + a\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 5 \cdot 10^{+138}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c + a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+150}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -1e+151) t_1 (if (<= t_2 1e+150) (+ (* i y) (+ a t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -1e+151) {
		tmp = t_1;
	} else if (t_2 <= 1e+150) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * log(c)
    t_2 = (b - 0.5d0) * log(c)
    if (t_2 <= (-1d+151)) then
        tmp = t_1
    else if (t_2 <= 1d+150) then
        tmp = (i * y) + (a + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * Math.log(c);
	double t_2 = (b - 0.5) * Math.log(c);
	double tmp;
	if (t_2 <= -1e+151) {
		tmp = t_1;
	} else if (t_2 <= 1e+150) {
		tmp = (i * y) + (a + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = b * math.log(c)
	t_2 = (b - 0.5) * math.log(c)
	tmp = 0
	if t_2 <= -1e+151:
		tmp = t_1
	elif t_2 <= 1e+150:
		tmp = (i * y) + (a + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -1e+151)
		tmp = t_1;
	elseif (t_2 <= 1e+150)
		tmp = Float64(Float64(i * y) + Float64(a + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = b * log(c);
	t_2 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (t_2 <= -1e+151)
		tmp = t_1;
	elseif (t_2 <= 1e+150)
		tmp = (i * y) + (a + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+151], t$95$1, If[LessEqual[t$95$2, 1e+150], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1.00000000000000002e151 or 9.99999999999999981e149 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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. lower-*.f64 N/A

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

      2. lower-log.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -1.00000000000000002e151 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.99999999999999981e149

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -1 \cdot 10^{+151}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+150}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -8.5e+211)
   (* (+ (/ z x) (log y)) x)
   (if (<= x 1.06e+217)
     (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
     (+ (+ a t) (fma (- b 0.5) (log c) (fma (log y) x z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -8.5e+211) {
		tmp = ((z / x) + log(y)) * x;
	} else if (x <= 1.06e+217) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(log(y), x, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -8.5e+211)
		tmp = Float64(Float64(Float64(z / x) + log(y)) * x);
	elseif (x <= 1.06e+217)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(log(y), x, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -8.5e+211], N[(N[(N[(z / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.06e+217], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000091e211

   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 x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\frac{z}{x} + \log y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 74.5%

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

   if -8.50000000000000091e211 < x < 1.06e217

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 1.06e217 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(\log y, x, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

5. Applied rewrites 84.3%

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

Alternative 8: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -8.5e+211)
   (* (+ (/ z x) (log y)) x)
   (if (<= x 1.6e+226)
     (+ (fma (- b 0.5) (log c) (fma y i z)) (+ a t))
     (+ (* x (log y)) (+ a t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -8.5e+211) {
		tmp = ((z / x) + log(y)) * x;
	} else if (x <= 1.6e+226) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z)) + (a + t);
	} else {
		tmp = (x * log(y)) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -8.5e+211)
		tmp = Float64(Float64(Float64(z / x) + log(y)) * x);
	elseif (x <= 1.6e+226)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, z)) + Float64(a + t));
	else
		tmp = Float64(Float64(x * log(y)) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -8.5e+211], N[(N[(N[(z / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.6e+226], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000091e211

   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 x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\frac{z}{x} + \log y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 74.5%

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

   if -8.50000000000000091e211 < x < 1.59999999999999989e226

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 1.59999999999999989e226 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right) + \left(a + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + b, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -8.5e+211)
   (* (+ (/ z x) (log y)) x)
   (if (<= x 1.6e+226)
     (+ (fma (+ -0.5 b) (log c) (fma y i z)) a)
     (+ (* x (log y)) (+ a t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -8.5e+211) {
		tmp = ((z / x) + log(y)) * x;
	} else if (x <= 1.6e+226) {
		tmp = fma((-0.5 + b), log(c), fma(y, i, z)) + a;
	} else {
		tmp = (x * log(y)) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -8.5e+211)
		tmp = Float64(Float64(Float64(z / x) + log(y)) * x);
	elseif (x <= 1.6e+226)
		tmp = Float64(fma(Float64(-0.5 + b), log(c), fma(y, i, z)) + a);
	else
		tmp = Float64(Float64(x * log(y)) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -8.5e+211], N[(N[(N[(z / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.6e+226], N[(N[(N[(-0.5 + b), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000091e211

   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 x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\frac{z}{x} + \log y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 74.5%

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

   if -8.50000000000000091e211 < x < 1.59999999999999989e226

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.5%

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

   if 1.59999999999999989e226 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{z}{x} + \log y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + b, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + b, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.8e+213)
     t_1
     (if (<= x 1.6e+226)
       (+ (fma (+ -0.5 b) (log c) (fma y i z)) a)
       (+ t_1 (+ a t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2.8e+213) {
		tmp = t_1;
	} else if (x <= 1.6e+226) {
		tmp = fma((-0.5 + b), log(c), fma(y, i, z)) + a;
	} else {
		tmp = t_1 + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.8e+213)
		tmp = t_1;
	elseif (x <= 1.6e+226)
		tmp = Float64(fma(Float64(-0.5 + b), log(c), fma(y, i, z)) + a);
	else
		tmp = Float64(t_1 + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+213], t$95$1, If[LessEqual[x, 1.6e+226], N[(N[(N[(-0.5 + b), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(t$95$1 + N[(a + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999999e213

   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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -2.7999999999999999e213 < x < 1.59999999999999989e226

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.5%

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

   if 1.59999999999999989e226 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + b, \log c, \mathsf{fma}\left(y, i, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 4.5e+172) (fma (- b 0.5) (log c) (fma y i z)) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 4.5e+172) {
		tmp = fma((b - 0.5), log(c), fma(y, i, z));
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 4.5e+172)
		tmp = fma(Float64(b - 0.5), log(c), fma(y, i, z));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 4.5e+172], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + z), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.5000000000000002e172

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 61.0%

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

   if 4.5000000000000002e172 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 84.9%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.4e+154) (fma y i (* b (log c))) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.4e+154) {
		tmp = fma(y, i, (b * log(c)));
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.4e+154)
		tmp = fma(y, i, Float64(b * log(c)));
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.4e+154], N[(y * i + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.40000000000000015e154

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 37.2

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

   5. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 45.5

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

   10. Applied rewrites 45.5%

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

   if 2.40000000000000015e154 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 81.4%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(y, i, b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+211}:\\ \;\;\;\;\frac{z}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -3.6e+211) (* (/ z x) x) (+ (* i y) (+ a t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -3.6e+211) {
		tmp = (z / x) * x;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (z <= (-3.6d+211)) then
        tmp = (z / x) * x
    else
        tmp = (i * y) + (a + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -3.6e+211) {
		tmp = (z / x) * x;
	} else {
		tmp = (i * y) + (a + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= -3.6e+211:
		tmp = (z / x) * x
	else:
		tmp = (i * y) + (a + t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -3.6e+211)
		tmp = Float64(Float64(z / x) * x);
	else
		tmp = Float64(Float64(i * y) + Float64(a + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= -3.6e+211)
		tmp = (z / x) * x;
	else
		tmp = (i * y) + (a + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -3.6e+211], N[(N[(z / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * y), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.60000000000000003e211

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.6%

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

   if -3.60000000000000003e211 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+211}:\\ \;\;\;\;\frac{z}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;i \cdot y + \left(a + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.7% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ i \cdot y + \left(a + t\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 87.9

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

5. Applied rewrites 87.9%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 55.5%

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

9. Final simplification 55.5%

   \[\leadsto i \cdot y + \left(a + t\right) \]
10. Add Preprocessing

Alternative 15: 38.3% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ i \cdot y + a \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 87.9

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

5. Applied rewrites 87.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 72.8%

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

9. Taylor expanded in y around inf

   \[\leadsto i \cdot y + a \]
10. Step-by-step derivation

11. Applied rewrites 40.8%

    \[\leadsto y \cdot i + a \]

12. Final simplification 40.8%

    \[\leadsto i \cdot y + a \]
13. Add Preprocessing

Alternative 16: 23.5% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ i \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.9

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

5. Applied rewrites 25.9%

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

6. Final simplification 25.9%

   \[\leadsto i \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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)))