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

Percentage Accurate: 99.9% → 99.9%

Time: 14.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ y + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, x\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (+ y (fma z (- 1.0 (log t)) (fma b (+ a -0.5) x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return y + fma(z, (1.0 - log(t)), fma(b, (a + -0.5), x));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(y + fma(z, Float64(1.0 - log(t)), fma(b, Float64(a + -0.5), x)))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. log-rec N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. associate-+r+ N/A

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

5. Applied rewrites 99.8%

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

Alternative 2: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -4e+130)
     t_2
     (if (<= t_1 4e+104) (fma z (- 1.0 (log t)) (+ x y)) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = y + fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -4e+130) {
		tmp = t_2;
	} else if (t_1 <= 4e+104) {
		tmp = fma(z, (1.0 - log(t)), (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(y + fma(b, Float64(a + -0.5), x))
	tmp = 0.0
	if (t_1 <= -4e+130)
		tmp = t_2;
	elseif (t_1 <= 4e+104)
		tmp = fma(z, Float64(1.0 - log(t)), Float64(x + y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+130], t$95$2, If[LessEqual[t$95$1, 4e+104], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e130 or 4e104 < (*.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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -4.0000000000000002e130 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4e104

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -4 \cdot 10^{+130}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+96}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) (fma b (+ a -0.5) y))))
   (if (<= z -1.6e-5) t_1 (if (<= z 5.2e+96) (+ y (fma b (+ a -0.5) x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), fma(b, (a + -0.5), y));
	double tmp;
	if (z <= -1.6e-5) {
		tmp = t_1;
	} else if (z <= 5.2e+96) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), fma(b, Float64(a + -0.5), y))
	tmp = 0.0
	if (z <= -1.6e-5)
		tmp = t_1;
	elseif (z <= 5.2e+96)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e-5], t$95$1, If[LessEqual[z, 5.2e+96], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.59999999999999993e-5 or 5.2e96 < 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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.9%

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

   if -1.59999999999999993e-5 < z < 5.2e96

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -5e-80)
   (+ (- (+ x z) (* z (log t))) (* a b))
   (fma z (- 1.0 (log t)) (fma b (+ a -0.5) y))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -5e-80) {
		tmp = ((x + z) - (z * log(t))) + (a * b);
	} else {
		tmp = fma(z, (1.0 - log(t)), fma(b, (a + -0.5), y));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -5e-80)
		tmp = Float64(Float64(Float64(x + z) - Float64(z * log(t))) + Float64(a * b));
	else
		tmp = fma(z, Float64(1.0 - log(t)), fma(b, Float64(a + -0.5), y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-80], N[(N[(N[(x + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -5e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.0

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

   8. Applied rewrites 69.0%

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

   if -5e-80 < (+.f64 x y)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.2%

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

4. Final simplification 75.8%

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

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -2.55 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, y\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -2.55e+127)
     (fma z t_1 y)
     (if (<= z 1.5e+137) (+ y (fma b (+ a -0.5) x)) (fma z t_1 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -2.55e+127) {
		tmp = fma(z, t_1, y);
	} else if (z <= 1.5e+137) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, t_1, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -2.55e+127)
		tmp = fma(z, t_1, y);
	elseif (z <= 1.5e+137)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, t_1, x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.55e+127], N[(z * t$95$1 + y), $MachinePrecision], If[LessEqual[z, 1.5e+137], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.55000000000000019e127

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.5%

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

   if -2.55000000000000019e127 < z < 1.5e137

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 1.5e137 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.6%

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

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, x\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) x)))
   (if (<= z -7e+207)
     t_1
     (if (<= z 1.5e+137) (+ y (fma b (+ a -0.5) x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), x);
	double tmp;
	if (z <= -7e+207) {
		tmp = t_1;
	} else if (z <= 1.5e+137) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), x)
	tmp = 0.0
	if (z <= -7e+207)
		tmp = t_1;
	elseif (z <= 1.5e+137)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -7e+207], t$95$1, If[LessEqual[z, 1.5e+137], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000056e207 or 1.5e137 < z

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.1%

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

   if -7.00000000000000056e207 < z < 1.5e137

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+208}:\\ \;\;\;\;z - z \cdot \log t\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -z, z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e+208)
   (- z (* z (log t)))
   (if (<= z 1.5e+137) (+ y (fma b (+ a -0.5) x)) (fma (log t) (- z) z))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e+208) {
		tmp = z - (z * log(t));
	} else if (z <= 1.5e+137) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(log(t), -z, z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e+208)
		tmp = Float64(z - Float64(z * log(t)));
	elseif (z <= 1.5e+137)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(log(t), Float64(-z), z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e+208], N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+137], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * (-z) + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e208

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -1.3e208 < z < 1.5e137

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if 1.5e137 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. log-rec N/A

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

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

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

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

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

      10. neg-mul-1 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lower-neg.f64 66.1

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

   7. Applied rewrites 66.1%

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

Alternative 8: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* z (log t)))))
   (if (<= z -1.3e+208)
     t_1
     (if (<= z 1.5e+137) (+ y (fma b (+ a -0.5) x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (z * log(t));
	double tmp;
	if (z <= -1.3e+208) {
		tmp = t_1;
	} else if (z <= 1.5e+137) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(z * log(t)))
	tmp = 0.0
	if (z <= -1.3e+208)
		tmp = t_1;
	elseif (z <= 1.5e+137)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e+208], t$95$1, If[LessEqual[z, 1.5e+137], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e208 or 1.5e137 < z

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -1.3e208 < z < 1.5e137

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 9: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+123}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* b (+ a -0.5))))
   (if (<= t_1 -2e+142) t_2 (if (<= t_1 1e+123) (+ x y) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -2e+142) {
		tmp = t_2;
	} else if (t_1 <= 1e+123) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    t_2 = b * (a + (-0.5d0))
    if (t_1 <= (-2d+142)) then
        tmp = t_2
    else if (t_1 <= 1d+123) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -2e+142) {
		tmp = t_2;
	} else if (t_1 <= 1e+123) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	t_2 = b * (a + -0.5)
	tmp = 0
	if t_1 <= -2e+142:
		tmp = t_2
	elif t_1 <= 1e+123:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (t_1 <= -2e+142)
		tmp = t_2;
	elseif (t_1 <= 1e+123)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	t_2 = b * (a + -0.5);
	tmp = 0.0;
	if (t_1 <= -2e+142)
		tmp = t_2;
	elseif (t_1 <= 1e+123)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+142], t$95$2, If[LessEqual[t$95$1, 1e+123], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e142 or 9.99999999999999978e122 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -2.0000000000000001e142 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.99999999999999978e122

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+142}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+123}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+142}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+221}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+142) (* a b) (if (<= t_1 4e+221) (+ x y) (* a b)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+142) {
		tmp = a * b;
	} else if (t_1 <= 4e+221) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if (t_1 <= (-2d+142)) then
        tmp = a * b
    else if (t_1 <= 4d+221) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+142) {
		tmp = a * b;
	} else if (t_1 <= 4e+221) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -2e+142:
		tmp = a * b
	elif t_1 <= 4e+221:
		tmp = x + y
	else:
		tmp = a * b
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+142)
		tmp = Float64(a * b);
	elseif (t_1 <= 4e+221)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -2e+142)
		tmp = a * b;
	elseif (t_1 <= 4e+221)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+142], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 4e+221], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e142 or 4.0000000000000002e221 < (*.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 71.8

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

   5. Applied rewrites 71.8%

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

   if -2.0000000000000001e142 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.0000000000000002e221

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+142}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 4 \cdot 10^{+221}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;b \leq -4.9 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+121}:\\ \;\;\;\;a \cdot b + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (+ a -0.5))))
   (if (<= b -4.9e+199) t_1 (if (<= b 2.8e+121) (+ (* a b) (+ x y)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a + -0.5);
	double tmp;
	if (b <= -4.9e+199) {
		tmp = t_1;
	} else if (b <= 2.8e+121) {
		tmp = (a * b) + (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (b <= (-4.9d+199)) then
        tmp = t_1
    else if (b <= 2.8d+121) then
        tmp = (a * b) + (x + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
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 (b <= -4.9e+199) {
		tmp = t_1;
	} else if (b <= 2.8e+121) {
		tmp = (a * b) + (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (a + -0.5)
	tmp = 0
	if b <= -4.9e+199:
		tmp = t_1
	elif b <= 2.8e+121:
		tmp = (a * b) + (x + y)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (b <= -4.9e+199)
		tmp = t_1;
	elseif (b <= 2.8e+121)
		tmp = Float64(Float64(a * b) + Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a + -0.5);
	tmp = 0.0;
	if (b <= -4.9e+199)
		tmp = t_1;
	elseif (b <= 2.8e+121)
		tmp = (a * b) + (x + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.9e+199], t$95$1, If[LessEqual[b, 2.8e+121], N[(N[(a * b), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.89999999999999965e199 or 2.80000000000000006e121 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -4.89999999999999965e199 < b < 2.80000000000000006e121

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 66.1

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

   8. Applied rewrites 66.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.9 \cdot 10^{+199}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+121}:\\ \;\;\;\;a \cdot b + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 9.7× speedup

Math

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

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ y (fma b (+ a -0.5) x)))

C

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

Julia

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

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 75.8

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

5. Applied rewrites 75.8%

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

Alternative 13: 42.0% accurate, 31.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ x + y \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ x y))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x + y;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + y
end function

Java

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

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return x + y

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(x + y)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = x + y;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in b around 0

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

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

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

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

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

   5. *-rgt-identity N/A

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

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

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 63.5

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

5. Applied rewrites 63.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 39.3%

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

9. Final simplification 39.3%

   \[\leadsto x + y \]
10. 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 2024220 
(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)))