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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 12

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := 1 - \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(t\_2, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (- 1.0 (log t))))
   (if (<= t_1 -5e+169)
     (fma (- a 0.5) b (+ y x))
     (if (<= t_1 5e+155)
       (+ (fma -0.5 b x) (fma t_2 z y))
       (fma t_2 z (fma (- a 0.5) b y))))))

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 = 1.0 - log(t);
	double tmp;
	if (t_1 <= -5e+169) {
		tmp = fma((a - 0.5), b, (y + x));
	} else if (t_1 <= 5e+155) {
		tmp = fma(-0.5, b, x) + fma(t_2, z, y);
	} else {
		tmp = fma(t_2, z, fma((a - 0.5), b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(1.0 - log(t))
	tmp = 0.0
	if (t_1 <= -5e+169)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	elseif (t_1 <= 5e+155)
		tmp = Float64(fma(-0.5, b, x) + fma(t_2, z, y));
	else
		tmp = fma(t_2, z, fma(Float64(a - 0.5), b, 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]}, Block[{t$95$2 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+169], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+155], N[(N[(-0.5 * b + x), $MachinePrecision] + N[(t$95$2 * z + y), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000017e169

   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 95.9

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

   5. Applied rewrites 95.9%

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

   if -5.00000000000000017e169 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999999e155

   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 96.8%

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

   if 4.9999999999999999e155 < (*.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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-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

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

   5. Applied rewrites 94.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+169}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+169)
     t_2
     (if (<= t_1 1e+112) (+ (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 <= -5e+169) {
		tmp = t_2;
	} else if (t_1 <= 1e+112) {
		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 <= -5e+169)
		tmp = t_2;
	elseif (t_1 <= 1e+112)
		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, -5e+169], t$95$2, If[LessEqual[t$95$1, 1e+112], 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) < -5.00000000000000017e169 or 9.9999999999999993e111 < (*.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 94.6

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

   5. Applied rewrites 94.6%

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

   if -5.00000000000000017e169 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.9999999999999993e111

   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 96.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right) + \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+169}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+169)
     t_2
     (if (<= t_1 1e+112) (+ (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 <= -5e+169) {
		tmp = t_2;
	} else if (t_1 <= 1e+112) {
		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 <= -5e+169)
		tmp = t_2;
	elseif (t_1 <= 1e+112)
		tmp = Float64(fma(Float64(1.0 - log(t)), z, 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, -5e+169], t$95$2, If[LessEqual[t$95$1, 1e+112], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000017e169 or 9.9999999999999993e111 < (*.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 94.6

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

   5. Applied rewrites 94.6%

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

   if -5.00000000000000017e169 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.9999999999999993e111

   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

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 90.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z y)))
   (if (<= z -5.9e+190)
     t_1
     (if (<= z 6.5e+167) (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, y);
	double tmp;
	if (z <= -5.9e+190) {
		tmp = t_1;
	} else if (z <= 6.5e+167) {
		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, y)
	tmp = 0.0
	if (z <= -5.9e+190)
		tmp = t_1;
	elseif (z <= 6.5e+167)
		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 + y), $MachinePrecision]}, If[LessEqual[z, -5.9e+190], t$95$1, If[LessEqual[z, 6.5e+167], 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 < -5.89999999999999972e190 or 6.5e167 < 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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-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

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.9%

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

   if -5.89999999999999972e190 < z < 6.5e167

   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 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 6: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-\log t, z, z\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- (log t)) z z)))
   (if (<= z -6e+191) t_1 (if (<= z 7.2e+167) (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(-log(t), z, z);
	double tmp;
	if (z <= -6e+191) {
		tmp = t_1;
	} else if (z <= 7.2e+167) {
		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(-log(t)), z, z)
	tmp = 0.0
	if (z <= -6e+191)
		tmp = t_1;
	elseif (z <= 7.2e+167)
		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[Log[t], $MachinePrecision]) * z + z), $MachinePrecision]}, If[LessEqual[z, -6e+191], t$95$1, If[LessEqual[z, 7.2e+167], 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 < -5.9999999999999995e191 or 7.20000000000000049e167 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{z \cdot 1 - z \cdot \log t} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 71.5

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

   5. Applied rewrites 71.5%

      \[\leadsto \color{blue}{z - \log t \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.6%

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

   if -5.9999999999999995e191 < z < 7.20000000000000049e167

   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 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z - \log t \cdot z\\ \mathbf{if}\;z \leq -6 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* (log t) z))))
   (if (<= z -6e+191) t_1 (if (<= z 7.2e+167) (fma (- a 0.5) b (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (log(t) * z);
	double tmp;
	if (z <= -6e+191) {
		tmp = t_1;
	} else if (z <= 7.2e+167) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(log(t) * z))
	tmp = 0.0
	if (z <= -6e+191)
		tmp = t_1;
	elseif (z <= 7.2e+167)
		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[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e+191], t$95$1, If[LessEqual[z, 7.2e+167], 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 < -5.9999999999999995e191 or 7.20000000000000049e167 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{z \cdot 1 - z \cdot \log t} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -5.9999999999999995e191 < z < 7.20000000000000049e167

   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 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 8: 37.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100000000000:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- a 0.5) -100000000000.0)
   (* b a)
   (if (<= (- a 0.5) -0.4) (* -0.5 b) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a - 0.5) <= -100000000000.0) {
		tmp = b * a;
	} else if ((a - 0.5) <= -0.4) {
		tmp = -0.5 * b;
	} 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) :: tmp
    if ((a - 0.5d0) <= (-100000000000.0d0)) then
        tmp = b * a
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = (-0.5d0) * b
    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 tmp;
	if ((a - 0.5) <= -100000000000.0) {
		tmp = b * a;
	} else if ((a - 0.5) <= -0.4) {
		tmp = -0.5 * b;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a - 0.5) <= -100000000000.0:
		tmp = b * a
	elif (a - 0.5) <= -0.4:
		tmp = -0.5 * b
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a - 0.5) <= -100000000000.0)
		tmp = Float64(b * a);
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(-0.5 * b);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a - 0.5) <= -100000000000.0)
		tmp = b * a;
	elseif ((a - 0.5) <= -0.4)
		tmp = -0.5 * b;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -100000000000.0], N[(b * a), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(-0.5 * b), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e11 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. 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 48.8

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

   5. Applied rewrites 48.8%

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

   if -1e11 < (-.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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-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

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 25.9

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

   8. Applied rewrites 25.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 25.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100000000000:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 74.6

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

5. Applied rewrites 74.6%

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

Alternative 10: 58.4% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. log-rec N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. associate-+r+ N/A

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

   10. *-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

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 54.1%

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

Alternative 11: 38.5% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $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 x around 0

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

   1. *-commutative N/A

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

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

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

   3. log-rec N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. associate-+r+ N/A

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

   10. *-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

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 38.0

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

8. Applied rewrites 38.0%

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

Alternative 12: 26.2% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * a), $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 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 26.4

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

5. Applied rewrites 26.4%

   \[\leadsto \color{blue}{b \cdot a} \]
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 2024295 
(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)))