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

Percentage Accurate: 99.8% → 99.9%

Time: 12.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   4. lower-fma.f64 99.5

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval 99.5

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. lower-neg.f64 99.5

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

   17. lift-+.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. associate-+l+ N/A

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

   20. lower-+.f64 N/A

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

   21. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+97}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -4e+196)
     t_2
     (if (<= t_1 4e+97) (+ x (fma b -0.5 (fma z (- 1.0 (log t)) y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = y + fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -4e+196) {
		tmp = t_2;
	} else if (t_1 <= 4e+97) {
		tmp = x + fma(b, -0.5, fma(z, (1.0 - log(t)), y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(y + fma(b, Float64(a + -0.5), x))
	tmp = 0.0
	if (t_1 <= -4e+196)
		tmp = t_2;
	elseif (t_1 <= 4e+97)
		tmp = Float64(x + fma(b, -0.5, fma(z, Float64(1.0 - log(t)), y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+196], t$95$2, If[LessEqual[t$95$1, 4e+97], N[(x + N[(b * -0.5 + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -3.9999999999999998e196 or 4.0000000000000003e97 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -3.9999999999999998e196 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.0000000000000003e97

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. log-rec N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Applied rewrites 92.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -4 \cdot 10^{+196}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+97}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+97}:\\ \;\;\;\;x + \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -1e-12)
     (fma (+ a -0.5) b (+ x y))
     (if (<= t_1 4e+97)
       (+ x (fma z (- 1.0 (log t)) y))
       (+ y (fma b (+ a -0.5) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e-12) {
		tmp = fma((a + -0.5), b, (x + y));
	} else if (t_1 <= 4e+97) {
		tmp = x + fma(z, (1.0 - log(t)), y);
	} else {
		tmp = y + fma(b, (a + -0.5), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -1e-12)
		tmp = fma(Float64(a + -0.5), b, Float64(x + y));
	elseif (t_1 <= 4e+97)
		tmp = Float64(x + fma(z, Float64(1.0 - log(t)), y));
	else
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-12], N[(N[(a + -0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+97], N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -9.9999999999999998e-13

   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

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

      4. lower-fma.f64 99.9

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval 99.9

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lower-+.f64 N/A

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

      21. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -9.9999999999999998e-13 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.0000000000000003e97

   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 inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. mul-1-neg N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 4.0000000000000003e97 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+97}:\\ \;\;\;\;x + \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -1.35e+42)
   (+ y (fma b (+ a -0.5) x))
   (fma z (- 1.0 (log t)) (fma b (+ a -0.5) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -1.35e+42) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, (1.0 - log(t)), fma(b, (a + -0.5), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -1.35e+42)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, Float64(1.0 - log(t)), fma(b, Float64(a + -0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -1.35e+42], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.35e42

   1. Initial program 98.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.35e42 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 77.2%

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

Alternative 5: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) y)))
   (if (<= z -3.1e+75)
     t_1
     (if (<= z 9.5e+115) (fma (+ a -0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), y);
	double tmp;
	if (z <= -3.1e+75) {
		tmp = t_1;
	} else if (z <= 9.5e+115) {
		tmp = fma((a + -0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), y)
	tmp = 0.0
	if (z <= -3.1e+75)
		tmp = t_1;
	elseif (z <= 9.5e+115)
		tmp = fma(Float64(a + -0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -3.1e+75], t$95$1, If[LessEqual[z, 9.5e+115], N[(N[(a + -0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000001e75 or 9.4999999999999997e115 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 62.6%

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

   if -3.1000000000000001e75 < z < 9.4999999999999997e115

   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. 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

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

      4. lower-fma.f64 100.0

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval 100.0

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 100.0

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lower-+.f64 N/A

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

      21. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 6: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+237}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.3e+237) (- z (* (log t) z)) (fma (+ a -0.5) b (+ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.3e+237) {
		tmp = z - (log(t) * z);
	} else {
		tmp = fma((a + -0.5), b, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.3e+237)
		tmp = Float64(z - Float64(log(t) * z));
	else
		tmp = fma(Float64(a + -0.5), b, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.3e+237], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(a + -0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2999999999999998e237

   1. Initial program 92.6%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. lower-*.f64 N/A

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

      14. lower-log.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -4.2999999999999998e237 < z

   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

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

      4. lower-fma.f64 99.9

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval 99.9

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lower-+.f64 N/A

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

      21. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 83.7

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

   7. Applied rewrites 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+237}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(b, a + -0.5, y\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (fma b (+ a -0.5) y)))
   (if (<= t_1 -1e+107) t_2 (if (<= t_1 2e+80) (fma -0.5 b (+ x y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = fma(b, (a + -0.5), y);
	double tmp;
	if (t_1 <= -1e+107) {
		tmp = t_2;
	} else if (t_1 <= 2e+80) {
		tmp = fma(-0.5, b, (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = fma(b, Float64(a + -0.5), y)
	tmp = 0.0
	if (t_1 <= -1e+107)
		tmp = t_2;
	elseif (t_1 <= 2e+80)
		tmp = fma(-0.5, b, Float64(x + y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+107], t$95$2, If[LessEqual[t$95$1, 2e+80], N[(-0.5 * b + 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) < -9.9999999999999997e106 or 2e80 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.5%

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

   if -9.9999999999999997e106 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2e80

   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. +-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

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

      4. lower-fma.f64 99.8

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval 99.8

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lower-+.f64 N/A

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

      21. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \leq 0.0082:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -0.5) (* a b) (if (<= a 0.0082) (* b -0.5) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = a * b;
	} else if (a <= 0.0082) {
		tmp = b * -0.5;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.5d0)) then
        tmp = a * b
    else if (a <= 0.0082d0) then
        tmp = b * (-0.5d0)
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = a * b;
	} else if (a <= 0.0082) {
		tmp = b * -0.5;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -0.5:
		tmp = a * b
	elif a <= 0.0082:
		tmp = b * -0.5
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -0.5)
		tmp = Float64(a * b);
	elseif (a <= 0.0082)
		tmp = Float64(b * -0.5);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -0.5)
		tmp = a * b;
	elseif (a <= 0.0082)
		tmp = b * -0.5;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -0.5], N[(a * b), $MachinePrecision], If[LessEqual[a, 0.0082], N[(b * -0.5), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.5 or 0.00820000000000000069 < a

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.8

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

   5. Applied rewrites 50.8%

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

   if -0.5 < a < 0.00820000000000000069

   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 inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 19.9%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \leq 0.0082:\\ \;\;\;\;b \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   4. lower-fma.f64 99.5

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval 99.5

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. lower-neg.f64 99.5

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

   17. lift-+.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. associate-+l+ N/A

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

   20. lower-+.f64 N/A

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

   21. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in z around 0

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

   1. lower-+.f64 79.5

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

7. Applied rewrites 79.5%

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

Alternative 10: 79.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 79.5

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

5. Applied rewrites 79.5%

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

Alternative 11: 58.7% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. log-rec N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. log-rec N/A

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

   14. mul-1-neg N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 77.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 58.2%

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

9. Final simplification 58.2%

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

Alternative 12: 38.3% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f64 37.9

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

5. Applied rewrites 37.9%

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

6. Final simplification 37.9%

   \[\leadsto \left(a + -0.5\right) \cdot b \]
7. Add Preprocessing

Alternative 13: 26.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ a \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 29.3

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

5. Applied rewrites 29.3%

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

6. Final simplification 29.3%

   \[\leadsto a \cdot b \]
7. 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 2024221 
(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)))