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

Percentage Accurate: 99.8% → 99.9%

Time: 10.5s

Alternatives: 17

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

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

FPCore

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

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), (z + (y + x)))));
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * -0.5 + N[(a * b + N[((-z) * N[Log[t], $MachinePrecision] + N[(z + N[(y + x), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.9

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

   16. lift--.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

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

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

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. Add Preprocessing

Alternative 2: 46.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4.5 \cdot 10^{+307}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -4.5e+307)
     (* b a)
     (if (<= t_1 -1e-102)
       (fma -0.5 b x)
       (if (<= t_1 2e+302) (fma -0.5 b y) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -4.5e+307) {
		tmp = b * a;
	} else if (t_1 <= -1e-102) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 2e+302) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -4.5e+307)
		tmp = Float64(b * a);
	elseif (t_1 <= -1e-102)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 2e+302)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	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[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4.5e+307], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -1e-102], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $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)) < -4.50000000000000025e307 or 2.0000000000000002e302 < (+.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 92.1

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

   5. Applied rewrites 92.1%

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

   if -4.50000000000000025e307 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.99999999999999933e-103

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.6%

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

   if -9.99999999999999933e-103 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.0000000000000002e302

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 37.6%

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

Alternative 3: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+128} \lor \neg \left(t\_1 \leq 10^{+40}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -5e+128) (not (<= t_1 1e+40)))
     (fma (- a 0.5) b (+ y x))
     (+ (fma (- 1.0 (log t)) z x) y))))

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+128) || !(t_1 <= 1e+40)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, x) + y;
	}
	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+128) || !(t_1 <= 1e+40))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(fma(Float64(1.0 - log(t)), z, x) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+128], N[Not[LessEqual[t$95$1, 1e+40]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5e128 or 1.00000000000000003e40 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -5e128 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000003e40

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. associate--l+ N/A

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

       3. unsub-neg N/A

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

       4. +-commutative N/A

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

       5. unsub-neg N/A

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

       6. *-rgt-identity N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-+.f64 92.8

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

   11. Applied rewrites 92.8%

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

   13. Applied rewrites 92.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+128} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 10^{+40}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right) + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \lor \neg \left(a - 0.5 \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + \mathsf{fma}\left(-0.5, b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (- a 0.5) -4.0) (not (<= (- a 0.5) 4e+59)))
   (fma (- a 0.5) b (+ y x))
   (+ (fma (- 1.0 (log t)) z y) (fma -0.5 b x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a - 0.5) <= -4.0) || !((a - 0.5) <= 4e+59)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, y) + fma(-0.5, b, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -4.0) || !(Float64(a - 0.5) <= 4e+59))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(fma(Float64(1.0 - log(t)), z, y) + fma(-0.5, b, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -4.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 4e+59]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + N[(-0.5 * b + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4 or 3.99999999999999989e59 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 91.9

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

   5. Applied rewrites 91.9%

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

   if -4 < (-.f64 a #s(literal 1/2 binary64)) < 3.99999999999999989e59

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \lor \neg \left(a - 0.5 \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + \mathsf{fma}\left(-0.5, b, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;a - 0.5 \leq -4:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y\right) + \mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (- a 0.5) -4.0)
     (fma (- a 0.5) b (+ y x))
     (if (<= (- a 0.5) 5e+37)
       (+ (fma t_1 z y) (fma -0.5 b x))
       (fma t_1 z (fma (- a 0.5) b x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((a - 0.5) <= -4.0) {
		tmp = fma((a - 0.5), b, (y + x));
	} else if ((a - 0.5) <= 5e+37) {
		tmp = fma(t_1, z, y) + fma(-0.5, b, x);
	} else {
		tmp = fma(t_1, z, fma((a - 0.5), b, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -4.0)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	elseif (Float64(a - 0.5) <= 5e+37)
		tmp = Float64(fma(t_1, z, y) + fma(-0.5, b, x));
	else
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -4.0], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+37], N[(N[(t$95$1 * z + y), $MachinePrecision] + N[(-0.5 * b + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4

   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 93.3

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

   5. Applied rewrites 93.3%

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

   if -4 < (-.f64 a #s(literal 1/2 binary64)) < 4.99999999999999989e37

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if 4.99999999999999989e37 < (-.f64 a #s(literal 1/2 binary64))

   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-sign-sub-inv N/A

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 87.5%

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

Alternative 6: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -1e-102)
   (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) - (z * log(t))) <= -1e-102) {
		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(z * log(t))) <= -1e-102)
		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[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-102], 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))) < -9.99999999999999933e-103

   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. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      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 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 60.2%

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

   if -9.99999999999999933e-103 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      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 80.9

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

   7. Applied rewrites 80.9%

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

   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 58.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -1e-102)
     (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-102) {
		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-102)
		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-102], 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) < -9.99999999999999933e-103

   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-sign-sub-inv N/A

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   if -9.99999999999999933e-103 < (+.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-sign-sub-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. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 77.0%

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

Alternative 8: 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]

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. Add Preprocessing

Alternative 9: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -6.5e+174)
     (fma t_1 z y)
     (if (<= z 8.2e+119) (fma (- a 0.5) b (+ y x)) (fma t_1 z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -6.5e+174) {
		tmp = fma(t_1, z, y);
	} else if (z <= 8.2e+119) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma(t_1, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -6.5e+174)
		tmp = fma(t_1, z, y);
	elseif (z <= 8.2e+119)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(t_1, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+174], N[(t$95$1 * z + y), $MachinePrecision], If[LessEqual[z, 8.2e+119], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5000000000000001e174

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 31.3%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. associate--l+ N/A

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

       3. unsub-neg N/A

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

       4. +-commutative N/A

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

       5. unsub-neg N/A

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

       6. *-rgt-identity N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-+.f64 86.1

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

   11. Applied rewrites 86.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 63.9%

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

   if -6.5000000000000001e174 < z < 8.1999999999999994e119

   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 93.0

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

   5. Applied rewrites 93.0%

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

   if 8.1999999999999994e119 < 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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. associate--l+ N/A

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

       3. unsub-neg N/A

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

       4. +-commutative N/A

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

       5. unsub-neg N/A

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

       6. *-rgt-identity N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-+.f64 82.3

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

   11. Applied rewrites 82.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 72.2%

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

Alternative 10: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 8.2e+119) (fma (- a 0.5) b (+ y x)) (fma (- 1.0 (log t)) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 8.2e+119) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 8.2e+119)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(Float64(1.0 - log(t)), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 8.2e+119], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.1999999999999994e119

   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 88.5

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

   5. Applied rewrites 88.5%

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

   if 8.1999999999999994e119 < 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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. associate--l+ N/A

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

       3. unsub-neg N/A

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

       4. +-commutative N/A

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

       5. unsub-neg N/A

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

       6. *-rgt-identity N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-+.f64 82.3

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

   11. Applied rewrites 82.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 72.2%

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

Alternative 11: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 2.3e+180) (fma (- a 0.5) b (+ y x)) (* (- 1.0 (log t)) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 2.3e+180) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = (1.0 - log(t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 2.3e+180)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(Float64(1.0 - log(t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 2.3e+180], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.2999999999999999e180

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 2.2999999999999999e180 < 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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 62.4

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

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+116} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+199}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -1e+116) (not (<= t_1 5e+199)))
     (fma (- a 0.5) b x)
     (fma -0.5 b (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -1e+116) || !(t_1 <= 5e+199)) {
		tmp = fma((a - 0.5), b, x);
	} else {
		tmp = fma(-0.5, b, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -1e+116) || !(t_1 <= 5e+199))
		tmp = fma(Float64(a - 0.5), b, x);
	else
		tmp = fma(-0.5, b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+116], N[Not[LessEqual[t$95$1, 5e+199]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.00000000000000002e116 or 4.9999999999999998e199 < (*.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. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      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 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 85.5%

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

   if -1.00000000000000002e116 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999998e199

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{+116} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 5 \cdot 10^{+199}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+128} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+229}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -5e+128) (not (<= t_1 2e+229))) t_1 (fma -0.5 b (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -5e+128) || !(t_1 <= 2e+229)) {
		tmp = t_1;
	} else {
		tmp = fma(-0.5, b, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -5e+128) || !(t_1 <= 2e+229))
		tmp = t_1;
	else
		tmp = fma(-0.5, b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+128], N[Not[LessEqual[t$95$1, 2e+229]], $MachinePrecision]], t$95$1, N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5e128 or 2e229 < (*.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. 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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.9

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

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

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

   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. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if -5e128 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2e229

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 68.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+128} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+229}\right):\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+31} \lor \neg \left(a \leq 2 \cdot 10^{+60}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.05e+31) (not (<= a 2e+60))) (* b a) (fma -0.5 b x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.05e+31) || !(a <= 2e+60)) {
		tmp = b * a;
	} else {
		tmp = fma(-0.5, b, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.05e+31) || !(a <= 2e+60))
		tmp = Float64(b * a);
	else
		tmp = fma(-0.5, b, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.05e+31], N[Not[LessEqual[a, 2e+60]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(-0.5 * b + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.04999999999999989e31 or 1.9999999999999999e60 < a

   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 66.5

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

   5. Applied rewrites 66.5%

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

   if -1.04999999999999989e31 < a < 1.9999999999999999e60

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 46.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+31} \lor \neg \left(a \leq 2 \cdot 10^{+60}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-7} \lor \neg \left(a \leq 3.7 \cdot 10^{+37}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -3.8e-7) (not (<= a 3.7e+37))) (* b a) (* -0.5 b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -3.8e-7) || !(a <= 3.7e+37)) {
		tmp = b * a;
	} else {
		tmp = -0.5 * 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) :: tmp
    if ((a <= (-3.8d-7)) .or. (.not. (a <= 3.7d+37))) then
        tmp = b * a
    else
        tmp = (-0.5d0) * 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 tmp;
	if ((a <= -3.8e-7) || !(a <= 3.7e+37)) {
		tmp = b * a;
	} else {
		tmp = -0.5 * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -3.8e-7) or not (a <= 3.7e+37):
		tmp = b * a
	else:
		tmp = -0.5 * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -3.8e-7) || !(a <= 3.7e+37))
		tmp = Float64(b * a);
	else
		tmp = Float64(-0.5 * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -3.8e-7) || ~((a <= 3.7e+37)))
		tmp = b * a;
	else
		tmp = -0.5 * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -3.8e-7], N[Not[LessEqual[a, 3.7e+37]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(-0.5 * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.80000000000000015e-7 or 3.6999999999999999e37 < a

   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 60.8

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

   5. Applied rewrites 60.8%

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

   if -3.80000000000000015e-7 < a < 3.6999999999999999e37

   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 a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 46.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 19.5%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-7} \lor \neg \left(a \leq 3.7 \cdot 10^{+37}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 81.2

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

5. Applied rewrites 81.2%

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

Alternative 17: 14.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

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

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

   3. log-rec N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. associate-+l+ N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. log-rec N/A

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

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

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

   14. *-commutative N/A

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

   15. lower-+.f64 N/A

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 53.5%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 32.8%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 11.9%

    \[\leadsto -0.5 \cdot b \]
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 2024313 
(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)))