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

Percentage Accurate: 99.8% → 99.9%

Time: 9.9s

Alternatives: 16

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.9

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. lower-fma.f64 N/A

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

   18. lower-neg.f64 99.9

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 51.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - \log t \cdot z\right) - \left(0.5 - a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (- (+ (+ x y) z) (* (log t) z)) (* (- 0.5 a) b))))
   (if (<= t_1 -1e-109)
     (fma (- a 0.5) b x)
     (if (<= t_1 4e+307) (fma -0.5 b y) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (log(t) * z)) - ((0.5 - a) * b);
	double tmp;
	if (t_1 <= -1e-109) {
		tmp = fma((a - 0.5), b, x);
	} else if (t_1 <= 4e+307) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(log(t) * z)) - Float64(Float64(0.5 - a) * b))
	tmp = 0.0
	if (t_1 <= -1e-109)
		tmp = fma(Float64(a - 0.5), b, x);
	elseif (t_1 <= 4e+307)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-109], N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+307], N[(-0.5 * b + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.9999999999999999e-110

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.3%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 61.2%

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

   if -9.9999999999999999e-110 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 3.99999999999999994e307

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 83.2

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.2%

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

   11. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, b, y\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 42.7%

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

   if 3.99999999999999994e307 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.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 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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - \log t \cdot z\right) - \left(0.5 - a\right) \cdot b \leq -1 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{elif}\;\left(\left(\left(x + y\right) + z\right) - \log t \cdot z\right) - \left(0.5 - a\right) \cdot b \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - \log t \cdot z \leq -1 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* (log t) z)) -1e-109)
   (fma (- a 0.5) b x)
   (fma (- a 0.5) b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + y) + z) - (log(t) * z)) <= -1e-109) {
		tmp = fma((a - 0.5), b, x);
	} else {
		tmp = fma((a - 0.5), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(log(t) * z)) <= -1e-109)
		tmp = fma(Float64(a - 0.5), b, x);
	else
		tmp = fma(Float64(a - 0.5), b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -1e-109], N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -9.9999999999999999e-110

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.5%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 61.8%

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

   if -9.9999999999999999e-110 < (-.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.0%

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

4. Final simplification 60.9%

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

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(-0.5, b, y\right)\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z (fma -0.5 b y))))
   (if (<= z -2.1e+195)
     t_1
     (if (<= z 5.8e+186) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, fma(-0.5, b, y));
	double tmp;
	if (z <= -2.1e+195) {
		tmp = t_1;
	} else if (z <= 5.8e+186) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, fma(-0.5, b, y))
	tmp = 0.0
	if (z <= -2.1e+195)
		tmp = t_1;
	elseif (z <= 5.8e+186)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(-0.5 * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+195], t$95$1, If[LessEqual[z, 5.8e+186], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000009e195 or 5.8e186 < 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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   if -2.10000000000000009e195 < z < 5.8e186

   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 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(-0.5, b, y\right)\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(-0.5, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -1e-109)
     (fma t_1 z (fma (- a 0.5) b x))
     (fma t_1 z (fma (- a 0.5) b y)))))

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) <= -1e-109) {
		tmp = fma(t_1, z, fma((a - 0.5), b, x));
	} else {
		tmp = fma(t_1, z, fma((a - 0.5), b, y));
	}
	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) <= -1e-109)
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, x));
	else
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, y));
	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], -1e-109], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.9999999999999999e-110

   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 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-sub-sign-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. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -9.9999999999999999e-110 < (+.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

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 77.2%

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

Alternative 6: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{a}\right) \cdot a, b, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 1e-6)
   (fma (- 1.0 (log t)) z (fma (- a 0.5) b x))
   (fma (* (- 1.0 (/ 0.5 a)) a) b (+ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 1e-6) {
		tmp = fma((1.0 - log(t)), z, fma((a - 0.5), b, x));
	} else {
		tmp = fma(((1.0 - (0.5 / a)) * a), b, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 1e-6)
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, x));
	else
		tmp = fma(Float64(Float64(1.0 - Float64(0.5 / a)) * a), b, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 1e-6], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 9.99999999999999955e-7

   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-sub-sign-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. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 9.99999999999999955e-7 < (+.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 89.3

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 89.3%

       \[\leadsto \mathsf{fma}\left(\left(1 - \frac{0.5}{a}\right) \cdot a, b, y + x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{a}\right) \cdot a, b, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.9

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. lower-fma.f64 N/A

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

   18. lower-neg.f64 99.9

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

6. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lift--.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-+l+ N/A

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (log(t) * z)) - ((0.5 - a) * 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) - (log(t) * z)) - ((0.5d0 - a) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $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. Final simplification 99.9%

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

Alternative 9: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z y)))
   (if (<= z -2.7e+195)
     t_1
     (if (<= z 1.1e+154) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, y);
	double tmp;
	if (z <= -2.7e+195) {
		tmp = t_1;
	} else if (z <= 1.1e+154) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, y)
	tmp = 0.0
	if (z <= -2.7e+195)
		tmp = t_1;
	elseif (z <= 1.1e+154)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision]}, If[LessEqual[z, -2.7e+195], t$95$1, If[LessEqual[z, 1.1e+154], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7000000000000002e195 or 1.1000000000000001e154 < 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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.7%

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

   if -2.7000000000000002e195 < z < 1.1000000000000001e154

   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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+241}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z x)))
   (if (<= z -1e+241)
     t_1
     (if (<= z 1.25e+148) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, x);
	double tmp;
	if (z <= -1e+241) {
		tmp = t_1;
	} else if (z <= 1.25e+148) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, x)
	tmp = 0.0
	if (z <= -1e+241)
		tmp = t_1;
	elseif (z <= 1.25e+148)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1e+241], t$95$1, If[LessEqual[z, 1.25e+148], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0000000000000001e241 or 1.25000000000000006e148 < 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 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-sub-sign-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. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   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 81.9%

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

   if -1.0000000000000001e241 < z < 1.25000000000000006e148

   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 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+241}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -1.05e+241)
     t_1
     (if (<= z 3.5e+231) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -1.05e+241) {
		tmp = t_1;
	} else if (z <= 3.5e+231) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -1.05e+241)
		tmp = t_1;
	elseif (z <= 3.5e+231)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.05e+241], t$95$1, If[LessEqual[z, 3.5e+231], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e241 or 3.4999999999999999e231 < 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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -1.05e241 < z < 3.4999999999999999e231

   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 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+241}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+288}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+288)
     (* a b)
     (if (<= t_1 -5e+141)
       (fma -0.5 b y)
       (if (<= t_1 1e+128)
         (+ x y)
         (if (<= t_1 2e+277) (fma -0.5 b y) (* a b)))))))

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 <= -5e+288) {
		tmp = a * b;
	} else if (t_1 <= -5e+141) {
		tmp = fma(-0.5, b, y);
	} else if (t_1 <= 1e+128) {
		tmp = x + y;
	} else if (t_1 <= 2e+277) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = a * b;
	}
	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 <= -5e+288)
		tmp = Float64(a * b);
	elseif (t_1 <= -5e+141)
		tmp = fma(-0.5, b, y);
	elseif (t_1 <= 1e+128)
		tmp = Float64(x + y);
	elseif (t_1 <= 2e+277)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(a * b);
	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[LessEqual[t$95$1, -5e+288], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, -5e+141], N[(-0.5 * b + y), $MachinePrecision], If[LessEqual[t$95$1, 1e+128], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+277], N[(-0.5 * b + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000003e288 or 2.00000000000000001e277 < (*.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 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 89.7

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

   5. Applied rewrites 89.7%

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

   if -5.0000000000000003e288 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000025e141 or 1.0000000000000001e128 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.00000000000000001e277

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 90.3

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.9%

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

   11. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, b, y\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 46.5%

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

   if -5.00000000000000025e141 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.0000000000000001e128

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 73.3

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.0%

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

   11. Taylor expanded in b around 0

       \[\leadsto x + \color{blue}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+288}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+138) t_1 (if (<= t_1 1e+128) (+ x y) t_1))))

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 <= -5e+138) {
		tmp = t_1;
	} else if (t_1 <= 1e+128) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-5d+138)) then
        tmp = t_1
    else if (t_1 <= 1d+128) then
        tmp = x + y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+138) {
		tmp = t_1;
	} else if (t_1 <= 1e+128) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -5e+138:
		tmp = t_1
	elif t_1 <= 1e+128:
		tmp = x + y
	else:
		tmp = t_1
	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 <= -5e+138)
		tmp = t_1;
	elseif (t_1 <= 1e+128)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+138)
		tmp = t_1;
	elseif (t_1 <= 1e+128)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+138], t$95$1, If[LessEqual[t$95$1, 1e+128], N[(x + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000016e138 or 1.0000000000000001e128 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 77.8%

       \[\leadsto \left(a - 0.5\right) \cdot b \]

   if -5.00000000000000016e138 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.0000000000000001e128

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.1%

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

   11. Taylor expanded in b around 0

       \[\leadsto x + \color{blue}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+205}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+138) (* a b) (if (<= t_1 1e+205) (+ x y) (* a b)))))

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 <= -5e+138) {
		tmp = a * b;
	} else if (t_1 <= 1e+205) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-5d+138)) then
        tmp = a * b
    else if (t_1 <= 1d+205) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static 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 <= -5e+138) {
		tmp = a * b;
	} else if (t_1 <= 1e+205) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -5e+138:
		tmp = a * b
	elif t_1 <= 1e+205:
		tmp = x + y
	else:
		tmp = a * b
	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 <= -5e+138)
		tmp = Float64(a * b);
	elseif (t_1 <= 1e+205)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+138)
		tmp = a * b;
	elseif (t_1 <= 1e+205)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+138], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+205], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000016e138 or 1.00000000000000002e205 < (*.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 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 55.7

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

   5. Applied rewrites 55.7%

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

   if -5.00000000000000016e138 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000002e205

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.6%

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

   11. Taylor expanded in b around 0

       \[\leadsto x + \color{blue}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.7%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+205}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $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 82.0

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

5. Applied rewrites 82.0%

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

6. Final simplification 82.0%

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

Alternative 16: 42.0% accurate, 31.5× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + y), $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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 82.0

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 60.3%

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

11. Taylor expanded in b around 0

    \[\leadsto x + \color{blue}{y} \]
12. Step-by-step derivation

13. Applied rewrites 41.7%

    \[\leadsto y + \color{blue}{x} \]

14. Final simplification 41.7%

    \[\leadsto x + y \]
15. 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 2024298 
(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)))