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

Percentage Accurate: 99.8% → 99.8%

Time: 11.7s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.9

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 88.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -100000000.0)
     t_2
     (if (<= t_1 1e+174) (+ (fma (- 1.0 (log t)) z x) y) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1e+174) {
		tmp = fma((1.0 - log(t)), z, x) + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_2;
	elseif (t_1 <= 1e+174)
		tmp = Float64(fma(Float64(1.0 - log(t)), z, x) + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -100000000.0], t$95$2, If[LessEqual[t$95$1, 1e+174], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision] + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e8 or 1.00000000000000007e174 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -1e8 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000007e174

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 96.7%

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

   9. Applied rewrites 96.7%

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

Alternative 3: 88.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -100000000.0)
     t_2
     (if (<= t_1 1e+174) (fma (- 1.0 (log t)) z (+ x y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1e+174) {
		tmp = fma((1.0 - log(t)), z, (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_2;
	elseif (t_1 <= 1e+174)
		tmp = fma(Float64(1.0 - log(t)), z, Float64(x + y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -100000000.0], t$95$2, If[LessEqual[t$95$1, 1e+174], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e8 or 1.00000000000000007e174 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -1e8 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000007e174

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -100000000:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+150}:\\ \;\;\;\;a \cdot b + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2e+150)
   (+ (* a b) (- (+ z (+ x y)) (* (log t) z)))
   (fma (- 1.0 (log t)) z (fma (- a 0.5) b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -2e+150) {
		tmp = (a * b) + ((z + (x + y)) - (log(t) * z));
	} else {
		tmp = fma((1.0 - log(t)), z, fma((a - 0.5), b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -2e+150)
		tmp = Float64(Float64(a * b) + Float64(Float64(z + Float64(x + y)) - Float64(log(t) * z)));
	else
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e+150], N[(N[(a * b), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.99999999999999996e150

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -1.99999999999999996e150 < (+.f64 x y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+150}:\\ \;\;\;\;a \cdot b + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ t_2 := \mathsf{fma}\left(1 - \log t, z, t\_1\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+119}:\\ \;\;\;\;t\_1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- a 0.5) b y)) (t_2 (fma (- 1.0 (log t)) z t_1)))
   (if (<= z -1.7e+107) t_2 (if (<= z 2.25e+119) (+ t_1 x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a - 0.5), b, y);
	double t_2 = fma((1.0 - log(t)), z, t_1);
	double tmp;
	if (z <= -1.7e+107) {
		tmp = t_2;
	} else if (z <= 2.25e+119) {
		tmp = t_1 + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a - 0.5), b, y)
	t_2 = fma(Float64(1.0 - log(t)), z, t_1)
	tmp = 0.0
	if (z <= -1.7e+107)
		tmp = t_2;
	elseif (z <= 2.25e+119)
		tmp = Float64(t_1 + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + t$95$1), $MachinePrecision]}, If[LessEqual[z, -1.7e+107], t$95$2, If[LessEqual[z, 2.25e+119], N[(t$95$1 + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6999999999999998e107 or 2.2500000000000001e119 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 93.3%

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

   if -1.6999999999999998e107 < z < 2.2500000000000001e119

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 6: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - \log t \cdot z \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ x y)) (* (log t) z)) -1.5e-8)
   (fma (- a 0.5) b x)
   (fma (- a 0.5) b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z + (x + y)) - (log(t) * z)) <= -1.5e-8) {
		tmp = fma((a - 0.5), b, x);
	} else {
		tmp = fma((a - 0.5), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(z + Float64(x + y)) - Float64(log(t) * z)) <= -1.5e-8)
		tmp = fma(Float64(a - 0.5), b, x);
	else
		tmp = fma(Float64(a - 0.5), b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -1.5e-8], N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -1.49999999999999987e-8

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.1%

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

   if -1.49999999999999987e-8 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-in N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 63.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - \log t \cdot z \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, a \cdot b\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z (* a b))))
   (if (<= z -3.8e+165)
     t_1
     (if (<= z 1.25e+148) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, (a * b));
	double tmp;
	if (z <= -3.8e+165) {
		tmp = t_1;
	} else if (z <= 1.25e+148) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, Float64(a * b))
	tmp = 0.0
	if (z <= -3.8e+165)
		tmp = t_1;
	elseif (z <= 1.25e+148)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e+165], t$95$1, If[LessEqual[z, 1.25e+148], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999999e165 or 1.25000000000000006e148 < z

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-in N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(1 - \log t, z, a \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 88.7%

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

   if -3.7999999999999999e165 < z < 1.25000000000000006e148

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

Alternative 8: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -5.2e+168)
     (fma t_1 z y)
     (if (<= z 2.4e+149) (+ (fma (- a 0.5) b y) x) (fma t_1 z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -5.2e+168) {
		tmp = fma(t_1, z, y);
	} else if (z <= 2.4e+149) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = fma(t_1, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -5.2e+168)
		tmp = fma(t_1, z, y);
	elseif (z <= 2.4e+149)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = fma(t_1, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+168], N[(t$95$1 * z + y), $MachinePrecision], If[LessEqual[z, 2.4e+149], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(t$95$1 * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e168

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-in N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 76.3%

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

   if -5.2e168 < z < 2.40000000000000012e149

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 2.40000000000000012e149 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 78.8%

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

Alternative 9: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+216}:\\ \;\;\;\;t\_1 \cdot z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -1.45e+216)
     (* t_1 z)
     (if (<= z 2.4e+149) (+ (fma (- a 0.5) b y) x) (fma t_1 z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -1.45e+216) {
		tmp = t_1 * z;
	} else if (z <= 2.4e+149) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = fma(t_1, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -1.45e+216)
		tmp = Float64(t_1 * z);
	elseif (z <= 2.4e+149)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = fma(t_1, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+216], N[(t$95$1 * z), $MachinePrecision], If[LessEqual[z, 2.4e+149], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(t$95$1 * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e216

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if -1.45e216 < z < 2.40000000000000012e149

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 2.40000000000000012e149 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 78.8%

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

Alternative 10: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+216}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;z - \log t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.45e+216)
   (* (- 1.0 (log t)) z)
   (if (<= z 2.5e+149) (+ (fma (- a 0.5) b y) x) (- z (* (log t) z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.45e+216) {
		tmp = (1.0 - log(t)) * z;
	} else if (z <= 2.5e+149) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = z - (log(t) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.45e+216)
		tmp = Float64(Float64(1.0 - log(t)) * z);
	elseif (z <= 2.5e+149)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(z - Float64(log(t) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.45e+216], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 2.5e+149], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e216

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if -1.45e216 < z < 2.49999999999999995e149

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 2.49999999999999995e149 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 72.0

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

   5. Applied rewrites 72.0%

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

Alternative 11: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z - \log t \cdot z\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* (log t) z))))
   (if (<= z -1.45e+216)
     t_1
     (if (<= z 2.5e+149) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (log(t) * z);
	double tmp;
	if (z <= -1.45e+216) {
		tmp = t_1;
	} else if (z <= 2.5e+149) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(log(t) * z))
	tmp = 0.0
	if (z <= -1.45e+216)
		tmp = t_1;
	elseif (z <= 2.5e+149)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+216], t$95$1, If[LessEqual[z, 2.5e+149], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e216 or 2.49999999999999995e149 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.45e216 < z < 2.49999999999999995e149

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 12: 68.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+176}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma (- a 0.5) b y)))
   (if (<= t_1 -5e+46) t_2 (if (<= t_1 1e+176) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y);
	double tmp;
	if (t_1 <= -5e+46) {
		tmp = t_2;
	} else if (t_1 <= 1e+176) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(Float64(a - 0.5), b, y)
	tmp = 0.0
	if (t_1 <= -5e+46)
		tmp = t_2;
	elseif (t_1 <= 1e+176)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+46], t$95$2, If[LessEqual[t$95$1, 1e+176], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000002e46 or 1e176 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-in N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 77.1%

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

   if -5.0000000000000002e46 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1e176

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 59.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+176}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+200}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+204) t_1 (if (<= t_1 1e+200) (+ x y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+204) {
		tmp = t_1;
	} else if (t_1 <= 1e+200) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-5d+204)) then
        tmp = t_1
    else if (t_1 <= 1d+200) then
        tmp = x + y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+204) {
		tmp = t_1;
	} else if (t_1 <= 1e+200) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -5e+204:
		tmp = t_1
	elif t_1 <= 1e+200:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -5e+204)
		tmp = t_1;
	elseif (t_1 <= 1e+200)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+204)
		tmp = t_1;
	elseif (t_1 <= 1e+200)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+204], t$95$1, If[LessEqual[t$95$1, 1e+200], N[(x + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000008e204 or 9.9999999999999997e199 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -5.00000000000000008e204 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.9999999999999997e199

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 54.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+200}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+215}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+200}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+215) (* a b) (if (<= t_1 1e+200) (+ x y) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+215) {
		tmp = a * b;
	} else if (t_1 <= 1e+200) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-5d+215)) then
        tmp = a * b
    else if (t_1 <= 1d+200) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+215) {
		tmp = a * b;
	} else if (t_1 <= 1e+200) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -5e+215:
		tmp = a * b
	elif t_1 <= 1e+200:
		tmp = x + y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -5e+215)
		tmp = Float64(a * b);
	elseif (t_1 <= 1e+200)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+215)
		tmp = a * b;
	elseif (t_1 <= 1e+200)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+215], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+200], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000001e215 or 9.9999999999999997e199 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -5.0000000000000001e215 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.9999999999999997e199

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 53.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+215}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+200}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 77.0

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

5. Applied rewrites 77.0%

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

Alternative 16: 42.1% accurate, 31.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. lift-+.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in b around 0

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

   1. associate-+r+ N/A

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

   2. *-lft-identity N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

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

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

   6. log-rec N/A

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

   7. distribute-rgt-in N/A

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

   8. log-rec N/A

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

   9. sub-neg N/A

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

   10. associate-+r+ N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. sub-neg N/A

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

   14. mul-1-neg N/A

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

   15. associate-+l+ N/A

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

   16. *-commutative N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f64 N/A

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

7. Applied rewrites 61.2%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 38.7%

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

11. Final simplification 38.7%

    \[\leadsto x + y \]
12. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + y) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + log(t)))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 1/2) b)))

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