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

Percentage Accurate: 99.8% → 99.9%

Time: 14.9s

Alternatives: 17

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-+r+ 99.5%

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

   6. +-commutative 99.5%

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

   7. +-commutative 99.5%

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

   8. *-commutative 99.5%

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

   9. cancel-sign-sub-inv 99.5%

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

   10. distribute-rgt1-in 99.5%

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

   11. *-commutative 99.5%

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

   12. fma-def 99.9%

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

   13. +-commutative 99.9%

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

   14. unsub-neg 99.9%

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

   15. fma-def 99.9%

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

   16. sub-neg 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -1e+94)
     (+ (+ x y) (* (+ a -0.5) b))
     (if (<= t_1 2e+51)
       (+ (+ x y) (* z (- 1.0 (log t))))
       (fma (+ a -0.5) b (+ x y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+94) {
		tmp = (x + y) + ((a + -0.5) * b);
	} else if (t_1 <= 2e+51) {
		tmp = (x + y) + (z * (1.0 - log(t)));
	} else {
		tmp = fma((a + -0.5), b, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -1e+94)
		tmp = Float64(Float64(x + y) + Float64(Float64(a + -0.5) * b));
	elseif (t_1 <= 2e+51)
		tmp = Float64(Float64(x + y) + Float64(z * Float64(1.0 - log(t))));
	else
		tmp = fma(Float64(a + -0.5), b, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+94], N[(N[(x + y), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(N[(x + y), $MachinePrecision] + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + -0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -1e94

   1. Initial program 98.3%

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

      1. associate--l+ 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 88.2%

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

   if -1e94 < (*.f64 (-.f64 a 1/2) b) < 2e51

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 97.8%

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

   if 2e51 < (*.f64 (-.f64 a 1/2) 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 87.6%

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

      1. fma-def 87.6%

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

      2. sub-neg 87.6%

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

      3. metadata-eval 87.6%

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

   6. Simplified 87.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \]

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq 4 \cdot 10^{+94}:\\ \;\;\;\;t_1 + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ a -0.5) b)))
   (if (<= (+ x y) 4e+94) (+ t_1 (- (+ x z) (* z (log t)))) (+ (+ x y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + -0.5) * b;
	double tmp;
	if ((x + y) <= 4e+94) {
		tmp = t_1 + ((x + z) - (z * log(t)));
	} else {
		tmp = (x + y) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + (-0.5d0)) * b
    if ((x + y) <= 4d+94) then
        tmp = t_1 + ((x + z) - (z * log(t)))
    else
        tmp = (x + y) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + -0.5) * b;
	double tmp;
	if ((x + y) <= 4e+94) {
		tmp = t_1 + ((x + z) - (z * Math.log(t)));
	} else {
		tmp = (x + y) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + -0.5) * b
	tmp = 0
	if (x + y) <= 4e+94:
		tmp = t_1 + ((x + z) - (z * math.log(t)))
	else:
		tmp = (x + y) + t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + -0.5) * b)
	tmp = 0.0
	if (Float64(x + y) <= 4e+94)
		tmp = Float64(t_1 + Float64(Float64(x + z) - Float64(z * log(t))));
	else
		tmp = Float64(Float64(x + y) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + -0.5) * b;
	tmp = 0.0;
	if ((x + y) <= 4e+94)
		tmp = t_1 + ((x + z) - (z * log(t)));
	else
		tmp = (x + y) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], 4e+94], N[(t$95$1 + N[(N[(x + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 4.0000000000000001e94

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 81.8%

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

      1. *-commutative 81.8%

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

   6. Simplified 81.8%

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

   if 4.0000000000000001e94 < (+.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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 90.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

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(x + y) + Float64(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[(x + y), $MachinePrecision] + N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+181}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- 1.0 (log t)))))
   (if (<= z -5.8e+259)
     t_1
     (if (<= z 7e+181)
       (+ (+ x y) (* (+ a -0.5) b))
       (if (<= z 5e+305) t_1 (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - log(t));
	double tmp;
	if (z <= -5.8e+259) {
		tmp = t_1;
	} else if (z <= 7e+181) {
		tmp = (x + y) + ((a + -0.5) * b);
	} else if (z <= 5e+305) {
		tmp = t_1;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (1.0d0 - log(t))
    if (z <= (-5.8d+259)) then
        tmp = t_1
    else if (z <= 7d+181) then
        tmp = (x + y) + ((a + (-0.5d0)) * b)
    else if (z <= 5d+305) then
        tmp = t_1
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - Math.log(t));
	double tmp;
	if (z <= -5.8e+259) {
		tmp = t_1;
	} else if (z <= 7e+181) {
		tmp = (x + y) + ((a + -0.5) * b);
	} else if (z <= 5e+305) {
		tmp = t_1;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z * (1.0 - math.log(t))
	tmp = 0
	if z <= -5.8e+259:
		tmp = t_1
	elif z <= 7e+181:
		tmp = (x + y) + ((a + -0.5) * b)
	elif z <= 5e+305:
		tmp = t_1
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(1.0 - log(t)))
	tmp = 0.0
	if (z <= -5.8e+259)
		tmp = t_1;
	elseif (z <= 7e+181)
		tmp = Float64(Float64(x + y) + Float64(Float64(a + -0.5) * b));
	elseif (z <= 5e+305)
		tmp = t_1;
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z * (1.0 - log(t));
	tmp = 0.0;
	if (z <= -5.8e+259)
		tmp = t_1;
	elseif (z <= 7e+181)
		tmp = (x + y) + ((a + -0.5) * b);
	elseif (z <= 5e+305)
		tmp = t_1;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.8e+259], t$95$1, If[LessEqual[z, 7e+181], N[(N[(x + y), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+305], t$95$1, N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7999999999999999e259 or 7.00000000000000016e181 < z < 5.00000000000000009e305

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-+r+ 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. cancel-sign-sub-inv 99.4%

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

      10. distribute-rgt1-in 99.8%

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

      11. *-commutative 99.8%

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

      12. fma-def 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

      15. fma-def 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 88.2%

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

   5. Taylor expanded in x around 0 75.3%

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

   6. Taylor expanded in z around inf 67.3%

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

   if -5.7999999999999999e259 < z < 7.00000000000000016e181

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 89.0%

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

   if 5.00000000000000009e305 < z

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+259}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+181}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+227} \lor \neg \left(z \leq 5.3 \cdot 10^{+181}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.4e+227) (not (<= z 5.3e+181)))
   (+ x (* z (- 1.0 (log t))))
   (+ (+ x y) (* (+ a -0.5) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.4e+227) || !(z <= 5.3e+181)) {
		tmp = x + (z * (1.0 - log(t)));
	} else {
		tmp = (x + y) + ((a + -0.5) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-4.4d+227)) .or. (.not. (z <= 5.3d+181))) then
        tmp = x + (z * (1.0d0 - log(t)))
    else
        tmp = (x + y) + ((a + (-0.5d0)) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.4e+227) || !(z <= 5.3e+181)) {
		tmp = x + (z * (1.0 - Math.log(t)));
	} else {
		tmp = (x + y) + ((a + -0.5) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -4.4e+227) or not (z <= 5.3e+181):
		tmp = x + (z * (1.0 - math.log(t)))
	else:
		tmp = (x + y) + ((a + -0.5) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.4e+227) || !(z <= 5.3e+181))
		tmp = Float64(x + Float64(z * Float64(1.0 - log(t))));
	else
		tmp = Float64(Float64(x + y) + Float64(Float64(a + -0.5) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -4.4e+227) || ~((z <= 5.3e+181)))
		tmp = x + (z * (1.0 - log(t)));
	else
		tmp = (x + y) + ((a + -0.5) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.4e+227], N[Not[LessEqual[z, 5.3e+181]], $MachinePrecision]], N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000004e227 or 5.2999999999999996e181 < z

   1. Initial program 97.2%

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

      1. associate--l+ 97.2%

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

      2. associate-+l+ 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+r+ 97.2%

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

      6. +-commutative 97.2%

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

      7. +-commutative 97.2%

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

      8. *-commutative 97.2%

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

      9. cancel-sign-sub-inv 97.2%

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

      10. distribute-rgt1-in 97.5%

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

      11. *-commutative 97.5%

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

      12. fma-def 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

      15. fma-def 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 76.5%

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

   if -4.4000000000000004e227 < z < 5.2999999999999996e181

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 89.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+227} \lor \neg \left(z \leq 5.3 \cdot 10^{+181}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 8: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+122} \lor \neg \left(z \leq 4.1 \cdot 10^{+181}\right):\\ \;\;\;\;y + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.5e+122) (not (<= z 4.1e+181)))
   (+ y (* z (- 1.0 (log t))))
   (+ (+ x y) (* (+ a -0.5) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.5e+122) || !(z <= 4.1e+181)) {
		tmp = y + (z * (1.0 - log(t)));
	} else {
		tmp = (x + y) + ((a + -0.5) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-2.5d+122)) .or. (.not. (z <= 4.1d+181))) then
        tmp = y + (z * (1.0d0 - log(t)))
    else
        tmp = (x + y) + ((a + (-0.5d0)) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.5e+122) || !(z <= 4.1e+181)) {
		tmp = y + (z * (1.0 - Math.log(t)));
	} else {
		tmp = (x + y) + ((a + -0.5) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.5e+122) or not (z <= 4.1e+181):
		tmp = y + (z * (1.0 - math.log(t)))
	else:
		tmp = (x + y) + ((a + -0.5) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.5e+122) || !(z <= 4.1e+181))
		tmp = Float64(y + Float64(z * Float64(1.0 - log(t))));
	else
		tmp = Float64(Float64(x + y) + Float64(Float64(a + -0.5) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.5e+122) || ~((z <= 4.1e+181)))
		tmp = y + (z * (1.0 - log(t)));
	else
		tmp = (x + y) + ((a + -0.5) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.5e+122], N[Not[LessEqual[z, 4.1e+181]], $MachinePrecision]], N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999994e122 or 4.10000000000000018e181 < z

   1. Initial program 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+r+ 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. *-commutative 98.2%

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

      9. cancel-sign-sub-inv 98.2%

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

      10. distribute-rgt1-in 98.4%

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

      11. *-commutative 98.4%

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

      12. fma-def 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

      15. fma-def 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 77.2%

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

   5. Taylor expanded in x around 0 63.3%

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

   if -2.49999999999999994e122 < z < 4.10000000000000018e181

   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. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 94.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+122} \lor \neg \left(z \leq 4.1 \cdot 10^{+181}\right):\\ \;\;\;\;y + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 9: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+122} \lor \neg \left(z \leq 4.9 \cdot 10^{+181}\right):\\ \;\;\;\;y + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.55e+122) (not (<= z 4.9e+181)))
   (+ y (* z (- 1.0 (log t))))
   (fma (+ a -0.5) b (+ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e+122) || !(z <= 4.9e+181)) {
		tmp = y + (z * (1.0 - log(t)));
	} else {
		tmp = fma((a + -0.5), b, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.55e+122) || !(z <= 4.9e+181))
		tmp = Float64(y + Float64(z * Float64(1.0 - log(t))));
	else
		tmp = fma(Float64(a + -0.5), b, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.55e+122], N[Not[LessEqual[z, 4.9e+181]], $MachinePrecision]], N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + -0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999999e122 or 4.89999999999999981e181 < z

   1. Initial program 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+r+ 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. *-commutative 98.2%

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

      9. cancel-sign-sub-inv 98.2%

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

      10. distribute-rgt1-in 98.4%

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

      11. *-commutative 98.4%

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

      12. fma-def 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

      15. fma-def 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 77.2%

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

   5. Taylor expanded in x around 0 63.3%

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

   if -1.54999999999999999e122 < z < 4.89999999999999981e181

   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. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. cancel-sign-sub-inv 100.0%

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

      10. distribute-rgt1-in 100.0%

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

      11. *-commutative 100.0%

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

      12. fma-def 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

      15. fma-def 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 94.1%

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

      1. fma-def 94.1%

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

      2. sub-neg 94.1%

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

      3. metadata-eval 94.1%

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

   6. Simplified 94.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+122} \lor \neg \left(z \leq 4.9 \cdot 10^{+181}\right):\\ \;\;\;\;y + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \]

Alternative 10: 64.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+178} \lor \neg \left(t_1 \leq 2 \cdot 10^{+145}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (or (<= t_1 -1e+178) (not (<= t_1 2e+145))) t_1 (+ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -1e+178) || !(t_1 <= 2e+145)) {
		tmp = t_1;
	} else {
		tmp = x + y;
	}
	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 <= (-1d+178)) .or. (.not. (t_1 <= 2d+145))) then
        tmp = t_1
    else
        tmp = x + y
    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 <= -1e+178) || !(t_1 <= 2e+145)) {
		tmp = t_1;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if (t_1 <= -1e+178) or not (t_1 <= 2e+145):
		tmp = t_1
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if ((t_1 <= -1e+178) || !(t_1 <= 2e+145))
		tmp = t_1;
	else
		tmp = Float64(x + y);
	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 <= -1e+178) || ~((t_1 <= 2e+145)))
		tmp = t_1;
	else
		tmp = x + y;
	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[Or[LessEqual[t$95$1, -1e+178], N[Not[LessEqual[t$95$1, 2e+145]], $MachinePrecision]], t$95$1, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -1.0000000000000001e178 or 2e145 < (*.f64 (-.f64 a 1/2) b)

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in z around 0 92.9%

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

   5. Taylor expanded in y around 0 87.1%

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

   6. Taylor expanded in b around inf 78.6%

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

   if -1.0000000000000001e178 < (*.f64 (-.f64 a 1/2) b) < 2e145

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 62.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+178} \lor \neg \left(b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+145}\right):\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11: 28.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-294}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e-183)
   x
   (if (<= y -5.9e-294)
     (* a b)
     (if (<= y 1.4e-279) x (if (<= y 1.45e+87) (* a b) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e-183) {
		tmp = x;
	} else if (y <= -5.9e-294) {
		tmp = a * b;
	} else if (y <= 1.4e-279) {
		tmp = x;
	} else if (y <= 1.45e+87) {
		tmp = a * b;
	} else {
		tmp = y;
	}
	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 (y <= (-2d-183)) then
        tmp = x
    else if (y <= (-5.9d-294)) then
        tmp = a * b
    else if (y <= 1.4d-279) then
        tmp = x
    else if (y <= 1.45d+87) then
        tmp = a * b
    else
        tmp = y
    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 (y <= -2e-183) {
		tmp = x;
	} else if (y <= -5.9e-294) {
		tmp = a * b;
	} else if (y <= 1.4e-279) {
		tmp = x;
	} else if (y <= 1.45e+87) {
		tmp = a * b;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2e-183:
		tmp = x
	elif y <= -5.9e-294:
		tmp = a * b
	elif y <= 1.4e-279:
		tmp = x
	elif y <= 1.45e+87:
		tmp = a * b
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e-183)
		tmp = x;
	elseif (y <= -5.9e-294)
		tmp = Float64(a * b);
	elseif (y <= 1.4e-279)
		tmp = x;
	elseif (y <= 1.45e+87)
		tmp = Float64(a * b);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2e-183)
		tmp = x;
	elseif (y <= -5.9e-294)
		tmp = a * b;
	elseif (y <= 1.4e-279)
		tmp = x;
	elseif (y <= 1.45e+87)
		tmp = a * b;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e-183], x, If[LessEqual[y, -5.9e-294], N[(a * b), $MachinePrecision], If[LessEqual[y, 1.4e-279], x, If[LessEqual[y, 1.45e+87], N[(a * b), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000001e-183 or -5.89999999999999994e-294 < y < 1.4e-279

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 100.0%

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

      11. *-commutative 100.0%

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

      12. fma-def 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

      15. fma-def 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 28.4%

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

   if -2.00000000000000001e-183 < y < -5.89999999999999994e-294 or 1.4e-279 < y < 1.4499999999999999e87

   1. Initial program 98.8%

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

      1. associate--l+ 98.8%

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

      2. sub-neg 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in z around 0 75.3%

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

   5. Taylor expanded in y around 0 71.9%

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

   6. Taylor expanded in a around inf 38.4%

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

      1. *-commutative 38.4%

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

   8. Simplified 38.4%

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

   if 1.4499999999999999e87 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 89.2%

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

   5. Taylor expanded in y around inf 55.9%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-294}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 12: 58.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-123}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5)))) (if (<= (+ x y) -2e-123) (+ x t_1) (+ y t_1))))

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 ((x + y) <= -2e-123) {
		tmp = x + t_1;
	} else {
		tmp = y + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if ((x + y) <= (-2d-123)) then
        tmp = x + t_1
    else
        tmp = y + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((x + y) <= -2e-123) {
		tmp = x + t_1;
	} else {
		tmp = y + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if (x + y) <= -2e-123:
		tmp = x + t_1
	else:
		tmp = y + t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (Float64(x + y) <= -2e-123)
		tmp = Float64(x + t_1);
	else
		tmp = Float64(y + t_1);
	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 ((x + y) <= -2e-123)
		tmp = x + t_1;
	else
		tmp = y + t_1;
	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[N[(x + y), $MachinePrecision], -2e-123], N[(x + t$95$1), $MachinePrecision], N[(y + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.0000000000000001e-123

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 57.3%

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

   if -2.0000000000000001e-123 < (+.f64 x y)

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 79.2%

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

   5. Taylor expanded in x around 0 59.1%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-123}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 13: 62.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+124}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.05e+124) (+ x (* b (- a 0.5))) (+ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.05e+124) {
		tmp = x + (b * (a - 0.5));
	} else {
		tmp = x + y;
	}
	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 (y <= 1.05d+124) then
        tmp = x + (b * (a - 0.5d0))
    else
        tmp = x + y
    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 (y <= 1.05e+124) {
		tmp = x + (b * (a - 0.5));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.05e+124:
		tmp = x + (b * (a - 0.5))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.05e+124)
		tmp = Float64(x + Float64(b * Float64(a - 0.5)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.05e+124)
		tmp = x + (b * (a - 0.5));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.05e+124], N[(x + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05000000000000006e124

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-+r+ 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. cancel-sign-sub-inv 99.4%

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

      10. distribute-rgt1-in 99.5%

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

      11. *-commutative 99.5%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 63.3%

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

   if 1.05000000000000006e124 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 61.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+124}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 14: 78.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in z around 0 79.6%

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

5. Final simplification 79.6%

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

Alternative 15: 51.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+92}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+74}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.1e+92) (* a b) (if (<= b 6e+74) (+ x y) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.1e+92) {
		tmp = a * b;
	} else if (b <= 6e+74) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.1d+92)) then
        tmp = a * b
    else if (b <= 6d+74) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.1e+92) {
		tmp = a * b;
	} else if (b <= 6e+74) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -2.1e+92:
		tmp = a * b
	elif b <= 6e+74:
		tmp = x + y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.1e+92)
		tmp = Float64(a * b);
	elseif (b <= 6e+74)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -2.1e+92)
		tmp = a * b;
	elseif (b <= 6e+74)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.1e+92], N[(a * b), $MachinePrecision], If[LessEqual[b, 6e+74], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.09999999999999986e92 or 6e74 < b

   1. Initial program 98.9%

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

      1. associate--l+ 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 87.9%

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

   5. Taylor expanded in y around 0 81.6%

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

   6. Taylor expanded in a around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

   if -2.09999999999999986e92 < b < 6e74

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 59.7%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+92}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+74}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 16: 27.5% accurate, 37.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= y 3.8e+66) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 3.8e+66) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 (y <= 3.8d+66) then
        tmp = x
    else
        tmp = y
    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 (y <= 3.8e+66) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 3.8e+66:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 3.8e+66)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 3.8e+66)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 3.8e+66], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000002e66

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-+r+ 99.4%

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

      6. +-commutative 99.4%

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

      7. +-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. cancel-sign-sub-inv 99.4%

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

      10. distribute-rgt1-in 99.4%

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

      11. *-commutative 99.4%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 26.2%

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

   if 3.8000000000000002e66 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 89.2%

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

   5. Taylor expanded in y around inf 55.9%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 17: 21.6% accurate, 115.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-+r+ 99.5%

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

   6. +-commutative 99.5%

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

   7. +-commutative 99.5%

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

   8. *-commutative 99.5%

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

   9. cancel-sign-sub-inv 99.5%

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

   10. distribute-rgt1-in 99.5%

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

   11. *-commutative 99.5%

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

   12. fma-def 99.9%

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

   13. +-commutative 99.9%

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

   14. unsub-neg 99.9%

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

   15. fma-def 99.9%

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

   16. sub-neg 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 22.7%

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

5. Final simplification 22.7%

   \[\leadsto x \]

Developer target: 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 2023171 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

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