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

Percentage Accurate: 99.9% → 99.9%

Time: 13.5s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. log-rec N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. associate-+r+ N/A

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

5. Simplified 99.9%

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

Alternative 2: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right) + \left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+225}:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b (- a 0.5)) (- (+ z (+ y x)) (* z (log t))))))
   (if (<= t_1 -5e+306)
     (* b a)
     (if (<= t_1 1e+225) (+ y (fma b -0.5 x)) (+ y (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * (a - 0.5)) + ((z + (y + x)) - (z * log(t)));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = b * a;
	} else if (t_1 <= 1e+225) {
		tmp = y + fma(b, -0.5, x);
	} else {
		tmp = y + (b * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * Float64(a - 0.5)) + Float64(Float64(z + Float64(y + x)) - Float64(z * log(t))))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(b * a);
	elseif (t_1 <= 1e+225)
		tmp = Float64(y + fma(b, -0.5, x));
	else
		tmp = Float64(y + Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+306], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+225], N[(y + N[(b * -0.5 + x), $MachinePrecision]), $MachinePrecision], N[(y + N[(b * a), $MachinePrecision]), $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)) < -4.99999999999999993e306

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -4.99999999999999993e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 9.99999999999999928e224

   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 inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in a around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 64.2

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

   11. Simplified 64.2%

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

   if 9.99999999999999928e224 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.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 x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) \leq -5 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) \leq 10^{+225}:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+237}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+68}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (fma b (+ a -0.5) x)))
   (if (<= t_1 -2e+237)
     t_2
     (if (<= t_1 4e+68)
       (+ x (fma b -0.5 (fma z (- 1.0 (log t)) y)))
       (+ y t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -2e+237) {
		tmp = t_2;
	} else if (t_1 <= 4e+68) {
		tmp = x + fma(b, -0.5, fma(z, (1.0 - log(t)), y));
	} else {
		tmp = y + t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = fma(b, Float64(a + -0.5), x)
	tmp = 0.0
	if (t_1 <= -2e+237)
		tmp = t_2;
	elseif (t_1 <= 4e+68)
		tmp = Float64(x + fma(b, -0.5, fma(z, Float64(1.0 - log(t)), y)));
	else
		tmp = Float64(y + t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+237], t$95$2, If[LessEqual[t$95$1, 4e+68], N[(x + N[(b * -0.5 + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999988e237

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-+.f64 100.0

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

   11. Simplified 100.0%

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

   if -1.99999999999999988e237 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 3.99999999999999981e68

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. log-rec N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Simplified 96.4%

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

   if 3.99999999999999981e68 < (*.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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 88.3

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

   5. Simplified 88.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+68}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+108}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -2e+108)
     t_2
     (if (<= t_1 4e+68) (fma z (- 1.0 (log t)) (+ y x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = y + fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -2e+108) {
		tmp = t_2;
	} else if (t_1 <= 4e+68) {
		tmp = fma(z, (1.0 - log(t)), (y + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(y + fma(b, Float64(a + -0.5), x))
	tmp = 0.0
	if (t_1 <= -2e+108)
		tmp = t_2;
	elseif (t_1 <= 4e+68)
		tmp = fma(z, Float64(1.0 - log(t)), Float64(y + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+108], t$95$2, If[LessEqual[t$95$1, 4e+68], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e108 or 3.99999999999999981e68 < (*.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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 90.0

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

   5. Simplified 90.0%

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

   if -2.0000000000000001e108 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 3.99999999999999981e68

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.2

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

   5. Simplified 95.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+108}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ y x)) (* z (log t))) -5e-187)
   (fma b (+ a -0.5) x)
   (fma b (+ a -0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z + (y + x)) - (z * log(t))) <= -5e-187) {
		tmp = fma(b, (a + -0.5), x);
	} else {
		tmp = fma(b, (a + -0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(z + Float64(y + x)) - Float64(z * log(t))) <= -5e-187)
		tmp = fma(b, Float64(a + -0.5), x);
	else
		tmp = fma(b, Float64(a + -0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-187], N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -4.9999999999999996e-187

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 80.1

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

   8. Simplified 80.1%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-+.f64 52.0

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

   11. Simplified 52.0%

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

   if -4.9999999999999996e-187 < (-.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. 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. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 73.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 53.8

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

   8. Simplified 53.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;y + x \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, \mathsf{fma}\left(z, t\_1, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ y x) -5e-187)
     (fma b (+ a -0.5) (fma z t_1 x))
     (fma z t_1 (fma b (+ a -0.5) 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 ((y + x) <= -5e-187) {
		tmp = fma(b, (a + -0.5), fma(z, t_1, x));
	} else {
		tmp = fma(z, t_1, fma(b, (a + -0.5), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(y + x) <= -5e-187)
		tmp = fma(b, Float64(a + -0.5), fma(z, t_1, x));
	else
		tmp = fma(z, t_1, fma(b, Float64(a + -0.5), 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[(y + x), $MachinePrecision], -5e-187], N[(b * N[(a + -0.5), $MachinePrecision] + N[(z * t$95$1 + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999996e-187

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 78.0

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

   8. Simplified 78.0%

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

   if -4.9999999999999996e-187 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   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. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 75.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, \mathsf{fma}\left(z, 1 - \log t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;y + x \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, \mathsf{fma}\left(z, t\_1, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, t\_1, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ y x) 2e+98)
     (fma b (+ a -0.5) (fma z t_1 x))
     (+ x (fma b -0.5 (fma z t_1 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 ((y + x) <= 2e+98) {
		tmp = fma(b, (a + -0.5), fma(z, t_1, x));
	} else {
		tmp = x + fma(b, -0.5, fma(z, t_1, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(y + x) <= 2e+98)
		tmp = fma(b, Float64(a + -0.5), fma(z, t_1, x));
	else
		tmp = Float64(x + fma(b, -0.5, fma(z, t_1, 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[(y + x), $MachinePrecision], 2e+98], N[(b * N[(a + -0.5), $MachinePrecision] + N[(z * t$95$1 + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(b * -0.5 + N[(z * t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 2e98

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 82.9

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

   8. Simplified 82.9%

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

   if 2e98 < (+.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 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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. log-rec N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Simplified 87.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, \mathsf{fma}\left(z, 1 - \log t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, x\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+204}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) x)))
   (if (<= z -1.5e+186)
     t_1
     (if (<= z 4.7e+204) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), x);
	double tmp;
	if (z <= -1.5e+186) {
		tmp = t_1;
	} else if (z <= 4.7e+204) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), x)
	tmp = 0.0
	if (z <= -1.5e+186)
		tmp = t_1;
	elseif (z <= 4.7e+204)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.5e+186], t$95$1, If[LessEqual[z, 4.7e+204], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999991e186 or 4.7000000000000002e204 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 86.8

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

   8. Simplified 86.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-log.f64 69.3

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

   11. Simplified 69.3%

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

   if -1.49999999999999991e186 < z < 4.7000000000000002e204

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 86.3

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

   5. Simplified 86.3%

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

Alternative 9: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-\log t, z, z\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+254}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- (log t)) z z)))
   (if (<= z -4.2e+189)
     t_1
     (if (<= z 4.8e+254) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-log(t), z, z);
	double tmp;
	if (z <= -4.2e+189) {
		tmp = t_1;
	} else if (z <= 4.8e+254) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-log(t)), z, z)
	tmp = 0.0
	if (z <= -4.2e+189)
		tmp = t_1;
	elseif (z <= 4.8e+254)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-N[Log[t], $MachinePrecision]) * z + z), $MachinePrecision]}, If[LessEqual[z, -4.2e+189], t$95$1, If[LessEqual[z, 4.8e+254], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999985e189 or 4.7999999999999997e254 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 73.1

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

   5. Simplified 73.1%

      \[\leadsto \color{blue}{z - z \cdot \log t} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 73.3

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

   7. Applied egg-rr 73.3%

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

   if -4.19999999999999985e189 < z < 4.7999999999999997e254

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 84.2

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

   5. Simplified 84.2%

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

Alternative 10: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+254}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* z (log t)))))
   (if (<= z -4.2e+189)
     t_1
     (if (<= z 4.8e+254) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (z * log(t));
	double tmp;
	if (z <= -4.2e+189) {
		tmp = t_1;
	} else if (z <= 4.8e+254) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(z * log(t)))
	tmp = 0.0
	if (z <= -4.2e+189)
		tmp = t_1;
	elseif (z <= 4.8e+254)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+189], t$95$1, If[LessEqual[z, 4.8e+254], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999985e189 or 4.7999999999999997e254 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 73.1

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

   5. Simplified 73.1%

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

   if -4.19999999999999985e189 < z < 4.7999999999999997e254

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 84.2

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

   5. Simplified 84.2%

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

Alternative 11: 57.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+214}:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -5e+306)
     (* b a)
     (if (<= t_1 -5e+214) (* b -0.5) (if (<= t_1 5e+211) (+ y x) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = b * a;
	} else if (t_1 <= -5e+214) {
		tmp = b * -0.5;
	} else if (t_1 <= 5e+211) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if (t_1 <= (-5d+306)) then
        tmp = b * a
    else if (t_1 <= (-5d+214)) then
        tmp = b * (-0.5d0)
    else if (t_1 <= 5d+211) then
        tmp = y + x
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = b * a;
	} else if (t_1 <= -5e+214) {
		tmp = b * -0.5;
	} else if (t_1 <= 5e+211) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -5e+306:
		tmp = b * a
	elif t_1 <= -5e+214:
		tmp = b * -0.5
	elif t_1 <= 5e+211:
		tmp = y + x
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(b * a);
	elseif (t_1 <= -5e+214)
		tmp = Float64(b * -0.5);
	elseif (t_1 <= 5e+211)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -5e+306)
		tmp = b * a;
	elseif (t_1 <= -5e+214)
		tmp = b * -0.5;
	elseif (t_1 <= 5e+211)
		tmp = y + x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+306], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -5e+214], N[(b * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+211], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.99999999999999993e306 or 4.9999999999999995e211 < (*.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 79.2

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

   5. Simplified 79.2%

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

   if -4.99999999999999993e306 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.99999999999999953e214

   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 b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 71.2%

      \[\leadsto b \cdot \color{blue}{-0.5} \]

   if -4.99999999999999953e214 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999995e211

   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 inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 56.8

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

   8. Simplified 56.8%

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

   9. Taylor expanded in b around 0

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

       1. lower-+.f64 57.5

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

   11. Simplified 57.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+214}:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+211}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+165}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* b (+ a -0.5))))
   (if (<= t_1 -5e+214) t_2 (if (<= t_1 2e+165) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -5e+214) {
		tmp = t_2;
	} else if (t_1 <= 2e+165) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    t_2 = b * (a + (-0.5d0))
    if (t_1 <= (-5d+214)) then
        tmp = t_2
    else if (t_1 <= 2d+165) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -5e+214) {
		tmp = t_2;
	} else if (t_1 <= 2e+165) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	t_2 = b * (a + -0.5)
	tmp = 0
	if t_1 <= -5e+214:
		tmp = t_2
	elif t_1 <= 2e+165:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (t_1 <= -5e+214)
		tmp = t_2;
	elseif (t_1 <= 2e+165)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	t_2 = b * (a + -0.5);
	tmp = 0.0;
	if (t_1 <= -5e+214)
		tmp = t_2;
	elseif (t_1 <= 2e+165)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+214], t$95$2, If[LessEqual[t$95$1, 2e+165], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.99999999999999953e214 or 1.9999999999999998e165 < (*.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 b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 82.6

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

   5. Simplified 82.6%

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

   if -4.99999999999999953e214 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.9999999999999998e165

   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 inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 76.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 55.4

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

   8. Simplified 55.4%

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

   9. Taylor expanded in b around 0

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

       1. lower-+.f64 58.4

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

   11. Simplified 58.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+214}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+165}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) 5e-136) (fma b (+ a -0.5) x) (+ y (* b a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= 5e-136) {
		tmp = fma(b, (a + -0.5), x);
	} else {
		tmp = y + (b * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y + x) <= 5e-136)
		tmp = fma(b, Float64(a + -0.5), x);
	else
		tmp = Float64(y + Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], 5e-136], N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 5.0000000000000002e-136

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 81.0

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

   8. Simplified 81.0%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-+.f64 53.9

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

   11. Simplified 53.9%

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

   if 5.0000000000000002e-136 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.6

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

   8. Simplified 44.6%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+170}:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+213}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e+170) (* b -0.5) (if (<= b 2e+213) (+ y x) (* b -0.5))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+170) {
		tmp = b * -0.5;
	} else if (b <= 2e+213) {
		tmp = y + x;
	} else {
		tmp = b * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-5d+170)) then
        tmp = b * (-0.5d0)
    else if (b <= 2d+213) then
        tmp = y + x
    else
        tmp = b * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+170) {
		tmp = b * -0.5;
	} else if (b <= 2e+213) {
		tmp = y + x;
	} else {
		tmp = b * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -5e+170:
		tmp = b * -0.5
	elif b <= 2e+213:
		tmp = y + x
	else:
		tmp = b * -0.5
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e+170)
		tmp = Float64(b * -0.5);
	elseif (b <= 2e+213)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -5e+170)
		tmp = b * -0.5;
	elseif (b <= 2e+213)
		tmp = y + x;
	else
		tmp = b * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e+170], N[(b * -0.5), $MachinePrecision], If[LessEqual[b, 2e+213], N[(y + x), $MachinePrecision], N[(b * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999977e170 or 1.99999999999999997e213 < 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 b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 86.3

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

   5. Simplified 86.3%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 44.2%

      \[\leadsto b \cdot \color{blue}{-0.5} \]

   if -4.99999999999999977e170 < b < 1.99999999999999997e213

   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 inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 61.0

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

   8. Simplified 61.0%

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

   9. Taylor expanded in b around 0

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

       1. lower-+.f64 50.8

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

   11. Simplified 50.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+170}:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+213}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-+l+ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 74.9

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

5. Simplified 74.9%

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

Alternative 16: 42.6% accurate, 31.5× speedup

Math

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

FPCore

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

C

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

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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Simplified 77.2%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 65.7

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

8. Simplified 65.7%

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

9. Taylor expanded in b around 0

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

    1. lower-+.f64 43.3

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

11. Simplified 43.3%

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

12. Final simplification 43.3%

    \[\leadsto y + x \]
13. 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 2024212 
(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)))