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

Percentage Accurate: 99.8% → 99.9%

Time: 14.5s

Alternatives: 19

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 99.9%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. +-lowering-+.f64 N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. log-lowering-log.f64 N/A

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

   12. neg-lowering-neg.f64 N/A

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

   13. associate-+l+ N/A

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

   14. +-lowering-+.f64 N/A

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

   15. +-lowering-+.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 egg-rr 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. Final simplification 99.9%

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

Alternative 2: 45.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b (- a 0.5)) (- (+ z (+ x y)) (* (log t) z)))))
   (if (<= t_1 (- INFINITY))
     (* a b)
     (if (<= t_1 -5e-58)
       (fma -0.5 b x)
       (if (<= t_1 5e+306) (fma -0.5 b y) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * (a - 0.5)) + ((z + (x + y)) - (log(t) * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = a * b;
	} else if (t_1 <= -5e-58) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 5e+306) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * Float64(a - 0.5)) + Float64(Float64(z + Float64(x + y)) - Float64(log(t) * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(a * b);
	elseif (t_1 <= -5e-58)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 5e+306)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, -5e-58], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], N[(-0.5 * b + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -inf.0 or 4.99999999999999993e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.99999999999999977e-58

   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 inf

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

   5. Simplified 51.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      2. accelerator-lowering-fma.f64 40.6

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

   8. Simplified 40.6%

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

   if -4.99999999999999977e-58 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 4.99999999999999993e306

   1. Initial program 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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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 y around inf

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

   7. Simplified 52.2%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 36.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b (- a 0.5)) (- (+ z (+ x y)) (* (log t) z)))))
   (if (<= t_1 (- INFINITY))
     (* a b)
     (if (<= t_1 -2e-71) (fma -0.5 b x) (fma a b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * (a - 0.5)) + ((z + (x + y)) - (log(t) * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = a * b;
	} else if (t_1 <= -2e-71) {
		tmp = fma(-0.5, b, x);
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * Float64(a - 0.5)) + Float64(Float64(z + Float64(x + y)) - Float64(log(t) * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(a * b);
	elseif (t_1 <= -2e-71)
		tmp = fma(-0.5, b, x);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, -2e-71], N[(-0.5 * b + x), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.9999999999999998e-71

   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 inf

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

   5. Simplified 51.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      2. accelerator-lowering-fma.f64 41.1

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

   8. Simplified 41.1%

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

   if -1.9999999999999998e-71 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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 y around inf

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

   7. Simplified 57.4%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 44.2%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+81}:\\ \;\;\;\;x + \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -1e+92)
     t_2
     (if (<= t_1 5e+81) (+ x (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 <= -1e+92) {
		tmp = t_2;
	} else if (t_1 <= 5e+81) {
		tmp = x + 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 <= -1e+92)
		tmp = t_2;
	elseif (t_1 <= 5e+81)
		tmp = Float64(x + 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, -1e+92], t$95$2, If[LessEqual[t$95$1, 5e+81], N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e92 or 4.9999999999999998e81 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. accelerator-lowering-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. +-lowering-+.f64 91.4

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

   5. Simplified 91.4%

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

   if -1e92 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999998e81

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 91.1

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

   5. Simplified 91.1%

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

      1. associate-+r+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 91.2

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

   7. Applied egg-rr 91.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+92}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+81}:\\ \;\;\;\;x + \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ t_2 := \mathsf{fma}\left(a, b, \mathsf{fma}\left(z, t\_1, x\right)\right)\\ \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a - 0.5 \leq 10^{+19}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, t\_1, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))) (t_2 (fma a b (fma z t_1 x))))
   (if (<= (- a 0.5) -4e+24)
     t_2
     (if (<= (- a 0.5) 1e+19) (+ x (fma b -0.5 (fma z t_1 y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double t_2 = fma(a, b, fma(z, t_1, x));
	double tmp;
	if ((a - 0.5) <= -4e+24) {
		tmp = t_2;
	} else if ((a - 0.5) <= 1e+19) {
		tmp = x + fma(b, -0.5, fma(z, t_1, y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	t_2 = fma(a, b, fma(z, t_1, x))
	tmp = 0.0
	if (Float64(a - 0.5) <= -4e+24)
		tmp = t_2;
	elseif (Float64(a - 0.5) <= 1e+19)
		tmp = Float64(x + fma(b, -0.5, fma(z, t_1, y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * b + N[(z * t$95$1 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+24], t$95$2, If[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+19], N[(x + N[(b * -0.5 + N[(z * t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -3.9999999999999999e24 or 1e19 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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 y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. log-lowering-log.f64 86.7

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

   7. Simplified 86.7%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 86.7%

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

   if -3.9999999999999999e24 < (-.f64 a #s(literal 1/2 binary64)) < 1e19

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Simplified 99.0%

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

Alternative 6: 21.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -2 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* b (- a 0.5)) (- (+ z (+ x y)) (* (log t) z))) -2e-71) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((b * (a - 0.5)) + ((z + (x + y)) - (log(t) * z))) <= -2e-71) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((b * (a - 0.5d0)) + ((z + (x + y)) - (log(t) * z))) <= (-2d-71)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((b * (a - 0.5)) + ((z + (x + y)) - (Math.log(t) * z))) <= -2e-71) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((b * (a - 0.5)) + ((z + (x + y)) - (math.log(t) * z))) <= -2e-71:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(b * Float64(a - 0.5)) + Float64(Float64(z + Float64(x + y)) - Float64(log(t) * z))) <= -2e-71)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((b * (a - 0.5)) + ((z + (x + y)) - (log(t) * z))) <= -2e-71)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-71], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.9999999999999998e-71

   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 inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 22.2%

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

   if -1.9999999999999998e-71 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 19.0%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(z + \left(x + y\right)\right) - \log t \cdot z\right) \leq -2 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - \log t \cdot z \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ x y)) (* (log t) z)) -5e-58)
   (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)) <= -5e-58) {
		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)) <= -5e-58)
		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], -5e-58], 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))) < -4.99999999999999977e-58

   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 inf

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

   5. Simplified 56.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 56.1

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

   7. Applied egg-rr 56.1%

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

   if -4.99999999999999977e-58 < (-.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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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 y around inf

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

   7. Simplified 58.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - \log t \cdot z \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+250}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -2.7e+95)
     (fma z t_1 x)
     (if (<= z 7e+250) (+ y (fma b (+ a -0.5) x)) (fma z t_1 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -2.7e+95) {
		tmp = fma(z, t_1, x);
	} else if (z <= 7e+250) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, t_1, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -2.7e+95)
		tmp = fma(z, t_1, x);
	elseif (z <= 7e+250)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, t_1, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.7e+95], N[(z * t$95$1 + x), $MachinePrecision], If[LessEqual[z, 7e+250], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e95

   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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 59.5

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

   8. Simplified 59.5%

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

   if -2.7e95 < z < 7.0000000000000001e250

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. accelerator-lowering-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. +-lowering-+.f64 86.3

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

   5. Simplified 86.3%

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

   if 7.0000000000000001e250 < 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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 71.2

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

   5. Simplified 71.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 71.2

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

   8. Simplified 71.2%

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

Alternative 9: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, x\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+231}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) x)))
   (if (<= z -2.3e+95)
     t_1
     (if (<= z 3.05e+231) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), x);
	double tmp;
	if (z <= -2.3e+95) {
		tmp = t_1;
	} else if (z <= 3.05e+231) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), x)
	tmp = 0.0
	if (z <= -2.3e+95)
		tmp = t_1;
	elseif (z <= 3.05e+231)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.3e+95], t$95$1, If[LessEqual[z, 3.05e+231], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999997e95 or 3.04999999999999986e231 < 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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 61.8

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

   8. Simplified 61.8%

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

   if -2.29999999999999997e95 < z < 3.04999999999999986e231

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. accelerator-lowering-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. +-lowering-+.f64 86.5

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

   5. Simplified 86.5%

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

Alternative 10: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+251}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- 1.0 (log t)))))
   (if (<= z -1.35e+180)
     t_1
     (if (<= z 1.2e+251) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - log(t));
	double tmp;
	if (z <= -1.35e+180) {
		tmp = t_1;
	} else if (z <= 1.2e+251) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(1.0 - log(t)))
	tmp = 0.0
	if (z <= -1.35e+180)
		tmp = t_1;
	elseif (z <= 1.2e+251)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.35e+180], t$95$1, If[LessEqual[z, 1.2e+251], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000008e180 or 1.19999999999999991e251 < 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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 74.8

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

   5. Simplified 74.8%

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

      1. associate-+r+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 74.8

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

   7. Applied egg-rr 74.8%

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

   8. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 68.0

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

   10. Simplified 68.0%

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

   if -1.35000000000000008e180 < z < 1.19999999999999991e251

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. accelerator-lowering-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. +-lowering-+.f64 83.7

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

   5. Simplified 83.7%

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

Alternative 11: 57.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+266}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -1e+266)
     (* a b)
     (if (<= t_1 -1e+92)
       (fma -0.5 b x)
       (if (<= t_1 5e+134) (+ x y) (* a b))))))

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+266) {
		tmp = a * b;
	} else if (t_1 <= -1e+92) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 5e+134) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	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+266)
		tmp = Float64(a * b);
	elseif (t_1 <= -1e+92)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 5e+134)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	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+266], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, -1e+92], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+134], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e266 or 4.99999999999999981e134 < (*.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 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. *-lowering-*.f64 66.1

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

   5. Simplified 66.1%

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

   if -1e266 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e92

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      2. accelerator-lowering-fma.f64 45.4

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

   8. Simplified 45.4%

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

   if -1e92 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999981e134

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 89.8

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

   5. Simplified 89.8%

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

   6. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 56.7

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

   8. Simplified 56.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+266}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* (+ a -0.5) b)))
   (if (<= t_1 -5e+131) t_2 (if (<= t_1 5e+134) (+ 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 = (a + -0.5) * b;
	double tmp;
	if (t_1 <= -5e+131) {
		tmp = t_2;
	} else if (t_1 <= 5e+134) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    t_2 = (a + (-0.5d0)) * b
    if (t_1 <= (-5d+131)) then
        tmp = t_2
    else if (t_1 <= 5d+134) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = (a + -0.5) * b;
	double tmp;
	if (t_1 <= -5e+131) {
		tmp = t_2;
	} else if (t_1 <= 5e+134) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	t_2 = (a + -0.5) * b
	tmp = 0
	if t_1 <= -5e+131:
		tmp = t_2
	elif t_1 <= 5e+134:
		tmp = 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 = Float64(Float64(a + -0.5) * b)
	tmp = 0.0
	if (t_1 <= -5e+131)
		tmp = t_2;
	elseif (t_1 <= 5e+134)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	t_2 = (a + -0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+131)
		tmp = t_2;
	elseif (t_1 <= 5e+134)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+131], t$95$2, If[LessEqual[t$95$1, 5e+134], 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) < -4.99999999999999995e131 or 4.99999999999999981e134 < (*.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. *-lowering-*.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. +-lowering-+.f64 79.5

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

   5. Simplified 79.5%

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

   if -4.99999999999999995e131 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999981e134

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 56.8

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

   8. Simplified 56.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+261}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+261) (* a b) (if (<= t_1 5e+134) (+ x y) (* a b)))))

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 <= -2e+261) {
		tmp = a * b;
	} else if (t_1 <= 5e+134) {
		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 = b * (a - 0.5d0)
    if (t_1 <= (-2d+261)) then
        tmp = a * b
    else if (t_1 <= 5d+134) 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 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+261) {
		tmp = a * b;
	} else if (t_1 <= 5e+134) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -2e+261:
		tmp = a * b
	elif t_1 <= 5e+134:
		tmp = x + y
	else:
		tmp = a * b
	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 <= -2e+261)
		tmp = Float64(a * b);
	elseif (t_1 <= 5e+134)
		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 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -2e+261)
		tmp = a * b;
	elseif (t_1 <= 5e+134)
		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[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+261], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 5e+134], 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) < -1.9999999999999999e261 or 4.99999999999999981e134 < (*.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 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. *-lowering-*.f64 64.4

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

   5. Simplified 64.4%

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

   if -1.9999999999999999e261 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999981e134

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. log-lowering-log.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 83.0

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

   5. Simplified 83.0%

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

   6. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 52.4

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

   8. Simplified 52.4%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+261}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -1e-22)
   (+ x (* a b))
   (if (<= (+ x y) 2e+114) (* (+ a -0.5) b) (fma a b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -1e-22) {
		tmp = x + (a * b);
	} else if ((x + y) <= 2e+114) {
		tmp = (a + -0.5) * b;
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -1e-22)
		tmp = Float64(x + Float64(a * b));
	elseif (Float64(x + y) <= 2e+114)
		tmp = Float64(Float64(a + -0.5) * b);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-22], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e+114], N[(N[(a + -0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1e-22

   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 inf

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

   5. Simplified 56.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 47.7

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

   8. Simplified 47.7%

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

   if -1e-22 < (+.f64 x y) < 2e114

   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. *-lowering-*.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. +-lowering-+.f64 48.7

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

   5. Simplified 48.7%

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

   if 2e114 < (+.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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 y around inf

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

   7. Simplified 63.6%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 53.1%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{if}\;a \leq -2.55 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.22 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (+ a -0.5) b x)))
   (if (<= a -2.55e+23) t_1 (if (<= a 1.22e+20) (fma -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((a + -0.5), b, x);
	double tmp;
	if (a <= -2.55e+23) {
		tmp = t_1;
	} else if (a <= 1.22e+20) {
		tmp = fma(-0.5, b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a + -0.5), b, x)
	tmp = 0.0
	if (a <= -2.55e+23)
		tmp = t_1;
	elseif (a <= 1.22e+20)
		tmp = fma(-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[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision]}, If[LessEqual[a, -2.55e+23], t$95$1, If[LessEqual[a, 1.22e+20], N[(-0.5 * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5500000000000001e23 or 1.22e20 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 70.1

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

   7. Applied egg-rr 70.1%

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

   if -2.5500000000000001e23 < a < 1.22e20

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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. +-lowering-+.f64 71.8

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

   7. Simplified 71.8%

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

   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. Simplified 70.9%

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

Alternative 16: 69.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot b\\ \mathbf{if}\;a \leq -6.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a b))))
   (if (<= a -6.4e+22) t_1 (if (<= a 8e+18) (fma -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 = x + (a * b);
	double tmp;
	if (a <= -6.4e+22) {
		tmp = t_1;
	} else if (a <= 8e+18) {
		tmp = fma(-0.5, b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * b))
	tmp = 0.0
	if (a <= -6.4e+22)
		tmp = t_1;
	elseif (a <= 8e+18)
		tmp = fma(-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[(x + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.4e+22], t$95$1, If[LessEqual[a, 8e+18], N[(-0.5 * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4e22 or 8e18 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.0%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

   if -6.4e22 < a < 8e18

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.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 egg-rr 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. +-lowering-+.f64 71.8

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

   7. Simplified 71.8%

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

   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. Simplified 70.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

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

   4. accelerator-lowering-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. +-lowering-+.f64 76.6

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

5. Simplified 76.6%

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

Alternative 18: 42.0% 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. Taylor expanded in b around 0

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

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

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

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

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

   5. *-rgt-identity N/A

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

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

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. --lowering--.f64 N/A

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

   14. log-lowering-log.f64 N/A

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

   15. +-commutative N/A

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

   16. +-lowering-+.f64 62.1

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

5. Simplified 62.1%

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

6. Taylor expanded in z around 0

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

   1. +-lowering-+.f64 39.6

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

8. Simplified 39.6%

   \[\leadsto \color{blue}{x + y} \]
9. Add Preprocessing

Alternative 19: 21.4% accurate, 126.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 19.9%

   \[\leadsto \color{blue}{x} \]
6. 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 2024205 
(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)))