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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 13

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: 99.9% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y + x\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (- 1.0 (log t)) z (fma (- a 0.5) b (+ y x))))

C

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

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, Float64(y + x)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+r+ N/A

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

   5. associate-+r- N/A

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

   6. *-rgt-identity N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. associate-+r+ N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+152} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+202}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + \mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e+152) (not (<= t_1 2e+202)))
     (fma (- a 0.5) b (+ y x))
     (+ (fma -0.5 b y) (fma (- 1.0 (log t)) z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -2e+152) || !(t_1 <= 2e+202)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma(-0.5, b, y) + fma((1.0 - log(t)), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -2e+152) || !(t_1 <= 2e+202))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(fma(-0.5, b, y) + fma(Float64(1.0 - log(t)), z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+152], N[Not[LessEqual[t$95$1, 2e+202]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * b + y), $MachinePrecision] + N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e152 or 1.9999999999999998e202 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. +-commutative N/A

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

      7. lower-+.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -2.0000000000000001e152 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.9999999999999998e202

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r- N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. associate-+r+ N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. log-rec N/A

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

      8. *-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   8. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, b, y\right) + \mathsf{fma}\left(1 - \log t, z, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+152} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+202}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + \mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+152} \lor \neg \left(t\_1 \leq 8 \cdot 10^{+108}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e+152) (not (<= t_1 8e+108)))
     (fma (- a 0.5) b (+ y x))
     (fma (- 1.0 (log t)) z (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -2e+152) || !(t_1 <= 8e+108)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -2e+152) || !(t_1 <= 8e+108))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+152], N[Not[LessEqual[t$95$1, 8e+108]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e152 or 8.0000000000000003e108 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. +-commutative N/A

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

      7. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -2.0000000000000001e152 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 8.0000000000000003e108

   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 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. +-commutative N/A

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

      7. lower-+.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. associate-+r+ N/A

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

      4. log-rec N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. log-rec N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      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. +-commutative N/A

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

      17. lower-+.f64 90.6

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

   8. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+152} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 8 \cdot 10^{+108}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+152} \lor \neg \left(t\_1 \leq 8 \cdot 10^{+108}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e+152) (not (<= t_1 8e+108)))
     (fma (- a 0.5) b (+ y x))
     (+ (fma (- 1.0 (log t)) z y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -2e+152) || !(t_1 <= 8e+108)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -2e+152) || !(t_1 <= 8e+108))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(fma(Float64(1.0 - log(t)), z, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+152], N[Not[LessEqual[t$95$1, 8e+108]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e152 or 8.0000000000000003e108 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. +-commutative N/A

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

      7. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -2.0000000000000001e152 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 8.0000000000000003e108

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r- N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. associate-+r+ N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 90.5

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

   8. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+152} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 8 \cdot 10^{+108}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -5e-47)
   (fma b (- a 0.5) x)
   (fma b (- a 0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + y) + z) - (z * log(t))) <= -5e-47) {
		tmp = fma(b, (a - 0.5), x);
	} else {
		tmp = fma(b, (a - 0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) <= -5e-47)
		tmp = fma(b, Float64(a - 0.5), x);
	else
		tmp = fma(b, Float64(a - 0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-47], N[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision], N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -5.00000000000000011e-47

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.2%

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

   if -5.00000000000000011e-47 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. +-commutative N/A

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

      7. lower-+.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.6%

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

Alternative 6: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + \mathsf{fma}\left(t\_1, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) 5e+112)
     (fma t_1 z (fma (- a 0.5) b x))
     (+ (fma -0.5 b y) (fma t_1 z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((x + y) <= 5e+112) {
		tmp = fma(t_1, z, fma((a - 0.5), b, x));
	} else {
		tmp = fma(-0.5, b, y) + fma(t_1, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(x + y) <= 5e+112)
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, x));
	else
		tmp = Float64(fma(-0.5, b, y) + fma(t_1, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], 5e+112], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * b + y), $MachinePrecision] + N[(t$95$1 * z + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 5e112

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 83.2%

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

   if 5e112 < (+.f64 x y)

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r- N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. associate-+r+ N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. log-rec N/A

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

      8. *-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   8. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, b, y\right) + \mathsf{fma}\left(1 - \log t, z, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+272} \lor \neg \left(z \leq 2.75 \cdot 10^{+166}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.2e+272) (not (<= z 2.75e+166)))
   (fma (- 1.0 (log t)) z x)
   (fma (- a 0.5) b (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.2e+272) || !(z <= 2.75e+166)) {
		tmp = fma((1.0 - log(t)), z, x);
	} else {
		tmp = fma((a - 0.5), b, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.2e+272) || !(z <= 2.75e+166))
		tmp = fma(Float64(1.0 - log(t)), z, x);
	else
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.2e+272], N[Not[LessEqual[z, 2.75e+166]], $MachinePrecision]], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999991e272 or 2.75000000000000004e166 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 74.7%

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

   if -4.19999999999999991e272 < z < 2.75000000000000004e166

   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. +-commutative N/A

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

      7. lower-+.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+272} \lor \neg \left(z \leq 2.75 \cdot 10^{+166}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z - \log t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 3.2e+166) (fma (- a 0.5) b (+ y x)) (- z (* (log t) z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 3.2e+166) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = z - (log(t) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 3.2e+166)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(z - Float64(log(t) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 3.2e+166], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999968e166

   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. +-commutative N/A

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

      7. lower-+.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 3.19999999999999968e166 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{z \cdot 1 - z \cdot \log t} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{z} - z \cdot \log t \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{z - z \cdot \log t} \]

      4. *-commutative N/A

         \[\leadsto z - \color{blue}{\log t \cdot z} \]

      5. lower-*.f64 N/A

         \[\leadsto z - \color{blue}{\log t \cdot z} \]

      6. lower-log.f64 71.8

         \[\leadsto z - \color{blue}{\log t} \cdot z \]

   5. Applied rewrites 71.8%

      \[\leadsto \color{blue}{z - \log t \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 37.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50 \lor \neg \left(a - 0.5 \leq 2000\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (- a 0.5) -50.0) (not (<= (- a 0.5) 2000.0))) (* b a) (* b -0.5)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a - 0.5) <= -50.0) || !((a - 0.5) <= 2000.0)) {
		tmp = b * a;
	} else {
		tmp = b * -0.5;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((a - 0.5d0) <= (-50.0d0)) .or. (.not. ((a - 0.5d0) <= 2000.0d0))) then
        tmp = b * a
    else
        tmp = b * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a - 0.5) <= -50.0) || !((a - 0.5) <= 2000.0)) {
		tmp = b * a;
	} else {
		tmp = b * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a - 0.5) <= -50.0) or not ((a - 0.5) <= 2000.0):
		tmp = b * a
	else:
		tmp = b * -0.5
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -50.0) || !(Float64(a - 0.5) <= 2000.0))
		tmp = Float64(b * a);
	else
		tmp = Float64(b * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a - 0.5) <= -50.0) || ~(((a - 0.5) <= 2000.0)))
		tmp = b * a;
	else
		tmp = b * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -50.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 2000.0]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(b * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -50 or 2e3 < (-.f64 a #s(literal 1/2 binary64))

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   if -50 < (-.f64 a #s(literal 1/2 binary64)) < 2e3

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r- N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. associate-+r+ N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 23.6

         \[\leadsto b \cdot \color{blue}{\left(a - 0.5\right)} \]

   8. Applied rewrites 23.6%

      \[\leadsto \color{blue}{b \cdot \left(a - 0.5\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto b \cdot \frac{-1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 22.8%

       \[\leadsto b \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50 \lor \neg \left(a - 0.5 \leq 2000\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $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. 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. +-commutative N/A

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

   7. lower-+.f64 78.8

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

5. Applied rewrites 78.8%

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

Alternative 11: 57.6% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $MachinePrecision] + x), $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. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

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

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

   3. log-rec N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. associate-+r+ N/A

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

   9. *-commutative N/A

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

   10. distribute-lft1-in N/A

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

   11. +-commutative N/A

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

   12. log-rec N/A

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

   13. mul-1-neg N/A

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

   14. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 59.3%

   \[\leadsto \mathsf{fma}\left(b, \color{blue}{a - 0.5}, x\right) \]
9. Add Preprocessing

Alternative 12: 37.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(a - 0.5\right) \end{array} \]

FPCore

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

C

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

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 = b * (a - 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $MachinePrecision]), $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. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+r+ N/A

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

   5. associate-+r- N/A

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

   6. *-rgt-identity N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. associate-+r+ N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 38.1

      \[\leadsto b \cdot \color{blue}{\left(a - 0.5\right)} \]

8. Applied rewrites 38.1%

   \[\leadsto \color{blue}{b \cdot \left(a - 0.5\right)} \]
9. Add Preprocessing

Alternative 13: 25.6% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

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 = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * a), $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. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.4

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

5. Applied rewrites 27.4%

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

Developer Target 1: 99.6% 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 2024324 
(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)))