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

Percentage Accurate: 99.8% → 99.8%

Time: 13.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. lift-+.f64 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(-z, \log t, \left(z + x\right) + y\right)\right)}}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 99.7

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

   \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(b, a - 0.5, \color{blue}{\mathsf{fma}\left(-\log t, z, y + x\right) + z}\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. remove-double-div 99.9

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

   4. lift-fma.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right) + z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+54)
     (+ (fma (- a 0.5) b (+ y x)) z)
     (if (<= t_1 2e+58)
       (fma (- 1.0 (log t)) z (+ y x))
       (+ (fma (- a 0.5) b x) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+54) {
		tmp = fma((a - 0.5), b, (y + x)) + z;
	} else if (t_1 <= 2e+58) {
		tmp = fma((1.0 - log(t)), z, (y + x));
	} else {
		tmp = fma((a - 0.5), b, x) + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+54)
		tmp = Float64(fma(Float64(a - 0.5), b, Float64(y + x)) + z);
	elseif (t_1 <= 2e+58)
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, x) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+54], N[(N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+58], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000002e54

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(-z, \log t, \left(z + x\right) + y\right)\right)}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(b, a - 0.5, \color{blue}{\mathsf{fma}\left(-\log t, z, y + x\right) + z}\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 99.9

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

      4. lift-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{x + y}\right) + z \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-+.f64 89.0

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

   11. Applied rewrites 89.0%

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

   if -2.0000000000000002e54 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999989e58

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-out-- N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. lower-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 1.99999999999999989e58 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right) + z\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right) + y\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = ((x + y) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + log(t)))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 1/2) b)))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))