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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 15

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

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (fma (- a 0.5) b (+ y x))))
   (if (<= t_1 -2e+95)
     t_2
     (if (<= t_1 5e+147) (+ (fma -0.5 b x) (fma (- 1.0 (log t)) z y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = fma((a - 0.5), b, (y + x));
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+147) {
		tmp = fma(-0.5, b, x) + fma((1.0 - log(t)), z, y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = fma(Float64(a - 0.5), b, Float64(y + x))
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+147)
		tmp = Float64(fma(-0.5, b, x) + fma(Float64(1.0 - log(t)), z, 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[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], t$95$2, If[LessEqual[t$95$1, 5e+147], N[(N[(-0.5 * b + x), $MachinePrecision] + N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000004e95 or 5.0000000000000002e147 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if -2.00000000000000004e95 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000002e147

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 92.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (fma (- a 0.5) b (+ y x))))
   (if (<= t_1 -2e+95)
     t_2
     (if (<= t_1 5e+60) (fma (- 1.0 (log t)) z (+ y x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = fma((a - 0.5), b, (y + x));
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+60) {
		tmp = fma((1.0 - log(t)), z, (y + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = fma(Float64(a - 0.5), b, Float64(y + x))
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+60)
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	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[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], t$95$2, If[LessEqual[t$95$1, 5e+60], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000004e95 or 4.99999999999999975e60 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -2.00000000000000004e95 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999975e60

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 91.6

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

   8. Applied rewrites 91.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;y + x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, b \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- a 0.5) b (+ y x))))
   (if (<= (+ y x) -5e+97)
     t_1
     (if (<= (+ y x) 5e+16) (fma (- 1.0 (log t)) z (* b (- a 0.5))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a - 0.5), b, (y + x));
	double tmp;
	if ((y + x) <= -5e+97) {
		tmp = t_1;
	} else if ((y + x) <= 5e+16) {
		tmp = fma((1.0 - log(t)), z, (b * (a - 0.5)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a - 0.5), b, Float64(y + x))
	tmp = 0.0
	if (Float64(y + x) <= -5e+97)
		tmp = t_1;
	elseif (Float64(y + x) <= 5e+16)
		tmp = fma(Float64(1.0 - log(t)), z, Float64(b * Float64(a - 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + x), $MachinePrecision], -5e+97], t$95$1, If[LessEqual[N[(y + x), $MachinePrecision], 5e+16], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.99999999999999999e97 or 5e16 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -4.99999999999999999e97 < (+.f64 x y) < 5e16

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, b \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - \log t \cdot z \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ y x)) (* (log t) z)) -5e-121)
   (fma b (- a 0.5) 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 + (y + x)) - (log(t) * z)) <= -5e-121) {
		tmp = fma(b, (a - 0.5), 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(y + x)) - Float64(log(t) * z)) <= -5e-121)
		tmp = fma(b, Float64(a - 0.5), 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[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -5e-121], N[(b * N[(a - 0.5), $MachinePrecision] + 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))) < -4.99999999999999989e-121

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.8%

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

   if -4.99999999999999989e-121 < (-.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. 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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - \log t \cdot z \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.25e+18)
   (fma (- 1.0 (log t)) z (fma (- a 0.5) b x))
   (fma (- a 0.5) b (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.25e+18) {
		tmp = fma((1.0 - log(t)), z, fma((a - 0.5), b, x));
	} else {
		tmp = fma((a - 0.5), b, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.25e+18)
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, x));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.25e+18], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e18

   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 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

   if 1.25e18 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z x)))
   (if (<= z -1.9e+208) t_1 (if (<= z 2e+185) (fma (- a 0.5) b (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, x);
	double tmp;
	if (z <= -1.9e+208) {
		tmp = t_1;
	} else if (z <= 2e+185) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, x)
	tmp = 0.0
	if (z <= -1.9e+208)
		tmp = t_1;
	elseif (z <= 2e+185)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.9e+208], t$95$1, If[LessEqual[z, 2e+185], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000001e208 or 2e185 < 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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. log-rec N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 83.5%

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

   if -1.9000000000000001e208 < z < 2e185

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 90.2

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

   5. Applied rewrites 90.2%

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

Alternative 8: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -3.1e+208)
     t_1
     (if (<= z 1.2e+257) (fma (- a 0.5) b (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -3.1e+208) {
		tmp = t_1;
	} else if (z <= 1.2e+257) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -3.1e+208)
		tmp = t_1;
	elseif (z <= 1.2e+257)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.1e+208], t$95$1, If[LessEqual[z, 1.2e+257], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.09999999999999981e208 or 1.2e257 < 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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -3.09999999999999981e208 < z < 1.2e257

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 9: 58.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+233}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+43}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -5e+233)
     (* b a)
     (if (<= t_1 5e+43) (+ y x) (if (<= t_1 4e+297) (fma b -0.5 x) (* 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+233) {
		tmp = b * a;
	} else if (t_1 <= 5e+43) {
		tmp = y + x;
	} else if (t_1 <= 4e+297) {
		tmp = fma(b, -0.5, x);
	} 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+233)
		tmp = Float64(b * a);
	elseif (t_1 <= 5e+43)
		tmp = Float64(y + x);
	elseif (t_1 <= 4e+297)
		tmp = fma(b, -0.5, x);
	else
		tmp = Float64(b * a);
	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, -5e+233], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 5e+43], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+297], N[(b * -0.5 + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000009e233 or 4.0000000000000001e297 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -5.00000000000000009e233 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000004e43

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 12.9%

       \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 58.5%

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

   if 5.0000000000000004e43 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.0000000000000001e297

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 50.6%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+233}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+43}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+149}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -5e+122) t_1 (if (<= t_1 1e+149) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+122) {
		tmp = t_1;
	} else if (t_1 <= 1e+149) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if (t_1 <= (-5d+122)) then
        tmp = t_1
    else if (t_1 <= 1d+149) then
        tmp = y + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -5e+122) {
		tmp = t_1;
	} else if (t_1 <= 1e+149) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -5e+122:
		tmp = t_1
	elif t_1 <= 1e+149:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -5e+122)
		tmp = t_1;
	elseif (t_1 <= 1e+149)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -5e+122)
		tmp = t_1;
	elseif (t_1 <= 1e+149)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+122], t$95$1, If[LessEqual[t$95$1, 1e+149], N[(y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.99999999999999989e122 or 1.00000000000000005e149 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 79.2%

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

   if -4.99999999999999989e122 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000005e149

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 72.4

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 62.3%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 9.7%

       \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 60.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+122}:\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+149}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+233}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+167}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -5e+233) (* b a) (if (<= t_1 1e+167) (+ y x) (* 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+233) {
		tmp = b * a;
	} else if (t_1 <= 1e+167) {
		tmp = y + x;
	} 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+233)) then
        tmp = b * a
    else if (t_1 <= 1d+167) then
        tmp = y + x
    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+233) {
		tmp = b * a;
	} else if (t_1 <= 1e+167) {
		tmp = y + x;
	} 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+233:
		tmp = b * a
	elif t_1 <= 1e+167:
		tmp = y + x
	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+233)
		tmp = Float64(b * a);
	elseif (t_1 <= 1e+167)
		tmp = Float64(y + x);
	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+233)
		tmp = b * a;
	elseif (t_1 <= 1e+167)
		tmp = y + x;
	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+233], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+167], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000009e233 or 1e167 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -5.00000000000000009e233 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1e167

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 73.8

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 62.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 14.9%

       \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 56.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+233}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+167}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) -5e+29) (+ y x) (if (<= (+ y x) 5e+28) (* -0.5 b) (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= -5e+29) {
		tmp = y + x;
	} else if ((y + x) <= 5e+28) {
		tmp = -0.5 * b;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y + x) <= (-5d+29)) then
        tmp = y + x
    else if ((y + x) <= 5d+28) then
        tmp = (-0.5d0) * b
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= -5e+29) {
		tmp = y + x;
	} else if ((y + x) <= 5e+28) {
		tmp = -0.5 * b;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y + x) <= -5e+29:
		tmp = y + x
	elif (y + x) <= 5e+28:
		tmp = -0.5 * b
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y + x) <= -5e+29)
		tmp = Float64(y + x);
	elseif (Float64(y + x) <= 5e+28)
		tmp = Float64(-0.5 * b);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y + x) <= -5e+29)
		tmp = y + x;
	elseif ((y + x) <= 5e+28)
		tmp = -0.5 * b;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], -5e+29], N[(y + x), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 5e+28], N[(-0.5 * b), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -5.0000000000000001e29 or 4.99999999999999957e28 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 25.3%

       \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 61.3%

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

   if -5.0000000000000001e29 < (+.f64 x y) < 4.99999999999999957e28

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. log-rec N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 26.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) 1e+46) (fma b (- a 0.5) x) (+ y x)))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], 1e+46], N[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 9.9999999999999999e45

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.0%

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

   if 9.9999999999999999e45 < (+.f64 x y)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 84.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 24.6%

       \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 67.4%

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

4. Final simplification 62.2%

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

Alternative 14: 78.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 80.6

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

5. Applied rewrites 80.6%

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

Alternative 15: 41.5% accurate, 31.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 80.6

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 68.7%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a - -0.5}, b, y + x\right) \]

8. Taylor expanded in b around inf

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

10. Applied rewrites 32.7%

    \[\leadsto \mathsf{fma}\left(a, a, -0.25\right) \cdot \color{blue}{\frac{b}{0.5 + a}} \]

11. Taylor expanded in b around 0

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

13. Applied rewrites 43.0%

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

Developer Target 1: 99.4% 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 2024296 
(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)))