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

Percentage Accurate: 99.8% → 99.8%

Time: 14.1s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

   6. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -1e+80)
     (+ y (fma b (+ a -0.5) x))
     (if (<= t_1 1e+28)
       (fma z (- 1.0 (log t)) (+ x y))
       (+ x (fma b (+ a -0.5) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+80) {
		tmp = y + fma(b, (a + -0.5), x);
	} else if (t_1 <= 1e+28) {
		tmp = fma(z, (1.0 - log(t)), (x + y));
	} else {
		tmp = x + fma(b, (a + -0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -1e+80)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	elseif (t_1 <= 1e+28)
		tmp = fma(z, Float64(1.0 - log(t)), Float64(x + y));
	else
		tmp = Float64(x + fma(b, Float64(a + -0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+80], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+28], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 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 91.9

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

   5. Simplified 91.9%

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

   if -1e80 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.99999999999999958e27

   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 97.2

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

   5. Simplified 97.2%

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

   if 9.99999999999999958e27 < (*.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. Step-by-step derivation

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 89.8

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

   7. Simplified 89.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+80}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a + -0.5, y\right)\\ \mathbf{if}\;b \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;x + t\_1\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) + b \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (+ a -0.5) y)))
   (if (<= b -2.7e+122)
     (+ x t_1)
     (if (<= b 1.7e+60)
       (+ (- (+ z (+ x y)) (* z (log t))) (* b a))
       (fma z (- 1.0 (log t)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, (a + -0.5), y);
	double tmp;
	if (b <= -2.7e+122) {
		tmp = x + t_1;
	} else if (b <= 1.7e+60) {
		tmp = ((z + (x + y)) - (z * log(t))) + (b * a);
	} else {
		tmp = fma(z, (1.0 - log(t)), t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, Float64(a + -0.5), y)
	tmp = 0.0
	if (b <= -2.7e+122)
		tmp = Float64(x + t_1);
	elseif (b <= 1.7e+60)
		tmp = Float64(Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))) + Float64(b * a));
	else
		tmp = fma(z, Float64(1.0 - log(t)), t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[b, -2.7e+122], N[(x + t$95$1), $MachinePrecision], If[LessEqual[b, 1.7e+60], N[(N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6999999999999998e122

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.1

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

   7. Simplified 98.1%

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

   if -2.6999999999999998e122 < b < 1.7e60

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.5

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

   5. Simplified 98.5%

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

   if 1.7e60 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 87.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a + -0.5, y\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) + b \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 21.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) + b \cdot \left(a - 0.5\right) \leq -4 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ z (+ x y)) (* z (log t))) (* b (- a 0.5))) -4e-128) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((z + (x + y)) - (z * log(t))) + (b * (a - 0.5))) <= -4e-128) {
		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 ((((z + (x + y)) - (z * log(t))) + (b * (a - 0.5d0))) <= (-4d-128)) 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 ((((z + (x + y)) - (z * Math.log(t))) + (b * (a - 0.5))) <= -4e-128) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (((z + (x + y)) - (z * math.log(t))) + (b * (a - 0.5))) <= -4e-128:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))) + Float64(b * Float64(a - 0.5))) <= -4e-128)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((((z + (x + y)) - (z * log(t))) + (b * (a - 0.5))) <= -4e-128)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-128], 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)) < -4.00000000000000022e-128

   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 22.4%

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

   if -4.00000000000000022e-128 < (+.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 23.3%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) + b \cdot \left(a - 0.5\right) \leq -4 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+155}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) (fma b (+ a -0.5) y))))
   (if (<= z -3.3e+86)
     t_1
     (if (<= z 8.4e+155) (+ 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)), fma(b, (a + -0.5), y));
	double tmp;
	if (z <= -3.3e+86) {
		tmp = t_1;
	} else if (z <= 8.4e+155) {
		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)), fma(b, Float64(a + -0.5), y))
	tmp = 0.0
	if (z <= -3.3e+86)
		tmp = t_1;
	elseif (z <= 8.4e+155)
		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] + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+86], t$95$1, If[LessEqual[z, 8.4e+155], 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 < -3.2999999999999999e86 or 8.4e155 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 91.3%

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

   if -3.2999999999999999e86 < z < 8.4e155

   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 95.0

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

   5. Simplified 95.0%

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

Alternative 6: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - z \cdot \log t \leq -4 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ x y)) (* z (log t))) -4e-128)
   (fma (+ a -0.5) b x)
   (fma b (+ a -0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z + (x + y)) - (z * log(t))) <= -4e-128) {
		tmp = fma((a + -0.5), b, x);
	} else {
		tmp = fma(b, (a + -0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))) <= -4e-128)
		tmp = fma(Float64(a + -0.5), b, x);
	else
		tmp = fma(b, Float64(a + -0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-128], N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -4.00000000000000022e-128

   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 56.2%

      \[\leadsto \color{blue}{x} + \left(a - 0.5\right) \cdot b \]
   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 56.2

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

   7. Applied egg-rr 56.2%

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

   if -4.00000000000000022e-128 < (-.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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 81.3

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

   7. Simplified 81.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 59.7

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

   10. Simplified 59.7%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - z \cdot \log t \leq -4 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;b \cdot a + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2e+84)
   (+ (* b a) (- (+ x z) (* z (log t))))
   (fma z (- 1.0 (log t)) (fma b (+ a -0.5) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -2e+84) {
		tmp = (b * a) + ((x + z) - (z * log(t)));
	} else {
		tmp = fma(z, (1.0 - log(t)), fma(b, (a + -0.5), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -2e+84)
		tmp = Float64(Float64(b * a) + Float64(Float64(x + z) - Float64(z * log(t))));
	else
		tmp = fma(z, Float64(1.0 - log(t)), fma(b, Float64(a + -0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e+84], N[(N[(b * a), $MachinePrecision] + N[(N[(x + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.00000000000000012e84

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 94.6

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

   5. Simplified 94.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 62.9

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

   8. Simplified 62.9%

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

   if -2.00000000000000012e84 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 82.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;b \cdot a + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((z + (x + y)) - (z * log(t))) + (b * (a - 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a - 0.5), $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. Final simplification 99.9%

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

Alternative 9: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+156}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) y)))
   (if (<= z -1.05e+182)
     t_1
     (if (<= z 1.32e+156) (+ 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)), y);
	double tmp;
	if (z <= -1.05e+182) {
		tmp = t_1;
	} else if (z <= 1.32e+156) {
		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)), y)
	tmp = 0.0
	if (z <= -1.05e+182)
		tmp = t_1;
	elseif (z <= 1.32e+156)
		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] + y), $MachinePrecision]}, If[LessEqual[z, -1.05e+182], t$95$1, If[LessEqual[z, 1.32e+156], 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 < -1.0499999999999999e182 or 1.3199999999999999e156 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 75.3

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

   5. Simplified 75.3%

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

   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 71.3

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

   8. Simplified 71.3%

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

   if -1.0499999999999999e182 < z < 1.3199999999999999e156

   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 92.2

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

   5. Simplified 92.2%

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

Alternative 10: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -z, z\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+205}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3e+216)
   (fma (log t) (- z) z)
   (if (<= z 3.6e+205) (+ y (fma b (+ a -0.5) x)) (fma z (- 1.0 (log t)) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3e+216) {
		tmp = fma(log(t), -z, z);
	} else if (z <= 3.6e+205) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, (1.0 - log(t)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3e+216)
		tmp = fma(log(t), Float64(-z), z);
	elseif (z <= 3.6e+205)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, Float64(1.0 - log(t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3e+216], N[(N[Log[t], $MachinePrecision] * (-z) + z), $MachinePrecision], If[LessEqual[z, 3.6e+205], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9999999999999998e216

   1. Initial program 99.4%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. neg-mul-1 N/A

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

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

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

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

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

      12. neg-mul-1 N/A

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

      13. neg-lowering-neg.f64 87.0

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

   7. Simplified 87.0%

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

   if -2.9999999999999998e216 < z < 3.60000000000000002e205

   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 87.5

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

   5. Simplified 87.5%

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

   if 3.60000000000000002e205 < z

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 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 81.2

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

   5. Simplified 81.2%

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

   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 76.8

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

   8. Simplified 76.8%

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

Alternative 11: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log t, -z, z\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+206}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (log t) (- z) z)))
   (if (<= z -1.8e+217)
     t_1
     (if (<= z 2.2e+206) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(log(t), -z, z);
	double tmp;
	if (z <= -1.8e+217) {
		tmp = t_1;
	} else if (z <= 2.2e+206) {
		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(log(t), Float64(-z), z)
	tmp = 0.0
	if (z <= -1.8e+217)
		tmp = t_1;
	elseif (z <= 2.2e+206)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * (-z) + z), $MachinePrecision]}, If[LessEqual[z, -1.8e+217], t$95$1, If[LessEqual[z, 2.2e+206], 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 < -1.8000000000000001e217 or 2.20000000000000001e206 < z

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. neg-mul-1 N/A

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

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

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

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

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

      12. neg-mul-1 N/A

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

      13. neg-lowering-neg.f64 81.3

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

   7. Simplified 81.3%

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

   if -1.8000000000000001e217 < z < 2.20000000000000001e206

   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 87.5

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

   5. Simplified 87.5%

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

Alternative 12: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+204}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* z (log t)))))
   (if (<= z -2.9e+217)
     t_1
     (if (<= z 4e+204) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (z * log(t));
	double tmp;
	if (z <= -2.9e+217) {
		tmp = t_1;
	} else if (z <= 4e+204) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(z * log(t)))
	tmp = 0.0
	if (z <= -2.9e+217)
		tmp = t_1;
	elseif (z <= 4e+204)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e+217], t$95$1, If[LessEqual[z, 4e+204], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999985e217 or 3.99999999999999996e204 < z

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

      14. log-lowering-log.f64 81.3

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

   5. Simplified 81.3%

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

   if -2.89999999999999985e217 < z < 3.99999999999999996e204

   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 87.5

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

   5. Simplified 87.5%

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

Alternative 13: 68.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* b (+ a -0.5))))
   (if (<= t_1 -5e+252) t_2 (if (<= t_1 5e+177) (+ x (fma b -0.5 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 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = t_2;
	} else if (t_1 <= 5e+177) {
		tmp = x + fma(b, -0.5, 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(b * Float64(a + -0.5))
	tmp = 0.0
	if (t_1 <= -5e+252)
		tmp = t_2;
	elseif (t_1 <= 5e+177)
		tmp = Float64(x + fma(b, -0.5, y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+252], t$95$2, If[LessEqual[t$95$1, 5e+177], N[(x + N[(b * -0.5 + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.9999999999999997e252 or 5.0000000000000003e177 < (*.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 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 91.3

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

   5. Simplified 91.3%

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

   if -4.9999999999999997e252 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000003e177

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 70.5

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

   7. Simplified 70.5%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 65.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+252}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+88}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* b (+ a -0.5))))
   (if (<= t_1 -5e+252) t_2 (if (<= t_1 2e+88) (+ 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 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = t_2;
	} else if (t_1 <= 2e+88) {
		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 = b * (a + (-0.5d0))
    if (t_1 <= (-5d+252)) then
        tmp = t_2
    else if (t_1 <= 2d+88) 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 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = t_2;
	} else if (t_1 <= 2e+88) {
		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 = b * (a + -0.5)
	tmp = 0
	if t_1 <= -5e+252:
		tmp = t_2
	elif t_1 <= 2e+88:
		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(b * Float64(a + -0.5))
	tmp = 0.0
	if (t_1 <= -5e+252)
		tmp = t_2;
	elseif (t_1 <= 2e+88)
		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 = b * (a + -0.5);
	tmp = 0.0;
	if (t_1 <= -5e+252)
		tmp = t_2;
	elseif (t_1 <= 2e+88)
		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[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+252], t$95$2, If[LessEqual[t$95$1, 2e+88], 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) < -4.9999999999999997e252 or 1.99999999999999992e88 < (*.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 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 84.2

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

   5. Simplified 84.2%

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

   if -4.9999999999999997e252 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999992e88

   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 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 61.4

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

   8. Simplified 61.4%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+252}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+88}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -5e+252) (* b a) (if (<= t_1 5e+177) (+ x y) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = b * a;
	} else if (t_1 <= 5e+177) {
		tmp = x + y;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if (t_1 <= (-5d+252)) then
        tmp = b * a
    else if (t_1 <= 5d+177) then
        tmp = x + y
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = b * a;
	} else if (t_1 <= 5e+177) {
		tmp = x + y;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -5e+252:
		tmp = b * a
	elif t_1 <= 5e+177:
		tmp = x + y
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -5e+252)
		tmp = Float64(b * a);
	elseif (t_1 <= 5e+177)
		tmp = Float64(x + y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -5e+252)
		tmp = b * a;
	elseif (t_1 <= 5e+177)
		tmp = x + y;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+252], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 5e+177], N[(x + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.9999999999999997e252 or 5.0000000000000003e177 < (*.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 72.1

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

   5. Simplified 72.1%

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

   if -4.9999999999999997e252 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000003e177

   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 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 59.0

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

   8. Simplified 59.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+252}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+177}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a, x\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- a 0.5) -1e+16)
   (+ y (fma b a x))
   (if (<= (- a 0.5) -0.4) (+ y (fma b -0.5 x)) (+ x (fma b a y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a - 0.5) <= -1e+16) {
		tmp = y + fma(b, a, x);
	} else if ((a - 0.5) <= -0.4) {
		tmp = y + fma(b, -0.5, x);
	} else {
		tmp = x + fma(b, a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a - 0.5) <= -1e+16)
		tmp = Float64(y + fma(b, a, x));
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(y + fma(b, -0.5, x));
	else
		tmp = Float64(x + fma(b, a, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+16], N[(y + N[(b * a + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(y + N[(b * -0.5 + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(b * a + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e16

   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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.3

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

   7. Simplified 83.3%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 83.3

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

   9. Applied egg-rr 83.3%

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

   10. Taylor expanded in a around inf

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

   12. Simplified 83.3%

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

   if -1e16 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 73.7

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

   9. Applied egg-rr 73.7%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 73.3

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

   12. Simplified 73.3%

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

   if -0.40000000000000002 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 81.1

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

   7. Simplified 81.1%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 81.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a, x\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(b, a, y\right)\\ \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma b a y))))
   (if (<= (- a 0.5) -1e+16)
     t_1
     (if (<= (- a 0.5) -0.4) (+ y (fma b -0.5 x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(b, a, y);
	double tmp;
	if ((a - 0.5) <= -1e+16) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4) {
		tmp = y + fma(b, -0.5, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(b, a, y))
	tmp = 0.0
	if (Float64(a - 0.5) <= -1e+16)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(y + fma(b, -0.5, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(b * a + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+16], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(y + N[(b * -0.5 + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e16 or -0.40000000000000002 < (-.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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 82.3

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

   7. Simplified 82.3%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 82.3%

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

   if -1e16 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 73.7

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

   9. Applied egg-rr 73.7%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 73.3

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

   12. Simplified 73.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;y + \mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(b, a, y\right)\\ \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma b a y))))
   (if (<= (- a 0.5) -1e+16)
     t_1
     (if (<= (- a 0.5) -0.4) (+ x (fma b -0.5 y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(b, a, y);
	double tmp;
	if ((a - 0.5) <= -1e+16) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4) {
		tmp = x + fma(b, -0.5, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(b, a, y))
	tmp = 0.0
	if (Float64(a - 0.5) <= -1e+16)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(x + fma(b, -0.5, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(b * a + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+16], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(x + N[(b * -0.5 + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e16 or -0.40000000000000002 < (-.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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 82.3

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

   7. Simplified 82.3%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 82.3%

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

   if -1e16 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 73.2%

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

Alternative 19: 49.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+85}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-24}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2e+85)
   (+ x (* b a))
   (if (<= (+ x y) 2e-24) (* b (+ a -0.5)) (+ y (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -2e+85) {
		tmp = x + (b * a);
	} else if ((x + y) <= 2e-24) {
		tmp = b * (a + -0.5);
	} else {
		tmp = y + (b * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x + y) <= (-2d+85)) then
        tmp = x + (b * a)
    else if ((x + y) <= 2d-24) then
        tmp = b * (a + (-0.5d0))
    else
        tmp = y + (b * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -2e+85) {
		tmp = x + (b * a);
	} else if ((x + y) <= 2e-24) {
		tmp = b * (a + -0.5);
	} else {
		tmp = y + (b * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x + y) <= -2e+85:
		tmp = x + (b * a)
	elif (x + y) <= 2e-24:
		tmp = b * (a + -0.5)
	else:
		tmp = y + (b * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -2e+85)
		tmp = Float64(x + Float64(b * a));
	elseif (Float64(x + y) <= 2e-24)
		tmp = Float64(b * Float64(a + -0.5));
	else
		tmp = Float64(y + Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x + y) <= -2e+85)
		tmp = x + (b * a);
	elseif ((x + y) <= 2e-24)
		tmp = b * (a + -0.5);
	else
		tmp = y + (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e+85], N[(x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-24], N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision], N[(y + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2e85

   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 54.0%

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

   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 49.5

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

   8. Simplified 49.5%

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

   if -2e85 < (+.f64 x y) < 1.99999999999999985e-24

   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 57.6

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

   5. Simplified 57.6%

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

   if 1.99999999999999985e-24 < (+.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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.2

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

   7. Simplified 83.2%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 83.2

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

   9. Applied egg-rr 83.2%

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

   10. Taylor expanded in a around inf

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

       1. *-lowering-*.f64 50.4

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

   12. Simplified 50.4%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+85}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-24}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-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 78.1

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

5. Simplified 78.1%

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

Alternative 21: 78.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

   6. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-+r+ N/A

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

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

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

   8. +-commutative N/A

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-lowering-+.f64 78.1

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

7. Simplified 78.1%

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

Alternative 22: 42.4% 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 64.0

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

5. Simplified 64.0%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 42.6

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

8. Simplified 42.6%

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

9. Final simplification 42.6%

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

Alternative 23: 22.3% 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 21.9%

   \[\leadsto \color{blue}{x} \]
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 2024204 
(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)))