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

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 17

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 45.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot b + x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b (- a 0.5)) (- (+ (+ y x) z) (* (log t) z)))))
   (if (<= t_1 (- INFINITY))
     (* b a)
     (if (<= t_1 -5e-104)
       (+ (* -0.5 b) x)
       (if (<= t_1 2e+306) (fma -0.5 b y) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * (a - 0.5)) + (((y + x) + z) - (log(t) * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = b * a;
	} else if (t_1 <= -5e-104) {
		tmp = (-0.5 * b) + x;
	} else if (t_1 <= 2e+306) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * Float64(a - 0.5)) + Float64(Float64(Float64(y + x) + z) - Float64(log(t) * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(b * a);
	elseif (t_1 <= -5e-104)
		tmp = Float64(Float64(-0.5 * b) + x);
	elseif (t_1 <= 2e+306)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -5e-104], N[(N[(-0.5 * b), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -inf.0 or 2.00000000000000003e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -inf.0 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.99999999999999979e-104

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 33.3%

      \[\leadsto -0.5 \cdot b + x \]

   if -4.99999999999999979e-104 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.00000000000000003e306

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 43.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right) \leq -\infty:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right) \leq -5 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot b + x\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := t\_1 + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ t_1 (- (+ (+ y x) z) (* (log t) z)))))
   (if (<= t_2 -5e-104) t_1 (if (<= t_2 2e+306) (fma -0.5 b y) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = t_1 + (((y + x) + z) - (log(t) * z));
	double tmp;
	if (t_2 <= -5e-104) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(t_1 + Float64(Float64(Float64(y + x) + z) - Float64(log(t) * z)))
	tmp = 0.0
	if (t_2 <= -5e-104)
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-104], t$95$1, If[LessEqual[t$95$2, 2e+306], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.99999999999999979e-104

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -4.99999999999999979e-104 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.00000000000000003e306

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 43.6%

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

   if 2.00000000000000003e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right) \leq -5 \cdot 10^{-104}:\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) + \left(\left(\left(y + x\right) + z\right) - \log t \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := 1 - \log t\\ t_3 := \mathsf{fma}\left(t\_2, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \mathsf{fma}\left(t\_2, z, y\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5)))
        (t_2 (- 1.0 (log t)))
        (t_3 (fma t_2 z (fma (- a 0.5) b y))))
   (if (<= t_1 -5e+172)
     t_3
     (if (<= t_1 5e+191) (+ (fma -0.5 b (fma t_2 z y)) x) t_3))))

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 t_3 = fma(t_2, z, fma((a - 0.5), b, y));
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = t_3;
	} else if (t_1 <= 5e+191) {
		tmp = fma(-0.5, b, fma(t_2, z, y)) + x;
	} else {
		tmp = t_3;
	}
	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))
	t_3 = fma(t_2, z, fma(Float64(a - 0.5), b, y))
	tmp = 0.0
	if (t_1 <= -5e+172)
		tmp = t_3;
	elseif (t_1 <= 5e+191)
		tmp = Float64(fma(-0.5, b, fma(t_2, z, y)) + x);
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(t$95$2 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+172], t$95$3, If[LessEqual[t$95$1, 5e+191], N[(N[(-0.5 * b + N[(t$95$2 * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000001e172 or 5.0000000000000002e191 < (*.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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   if -5.0000000000000001e172 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000002e191

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 94.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \mathsf{fma}\left(1 - \log t, z, y\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := 1 - \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \mathsf{fma}\left(t\_2, z, y\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

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+220)
     (fma t_2 z t_1)
     (if (<= t_1 1e+136)
       (+ (fma -0.5 b (fma t_2 z y)) x)
       (+ (fma (- a 0.5) b y) x)))))

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+220) {
		tmp = fma(t_2, z, t_1);
	} else if (t_1 <= 1e+136) {
		tmp = fma(-0.5, b, fma(t_2, z, y)) + x;
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	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+220)
		tmp = fma(t_2, z, t_1);
	elseif (t_1 <= 1e+136)
		tmp = Float64(fma(-0.5, b, fma(t_2, z, y)) + x);
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	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+220], N[(t$95$2 * z + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 1e+136], N[(N[(-0.5 * b + N[(t$95$2 * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000002e220

   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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.0%

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

   if -5.0000000000000002e220 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000006e136

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 95.3%

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

   if 1.00000000000000006e136 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, b \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \mathsf{fma}\left(1 - \log t, z, y\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := 1 - \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

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+172)
     (fma t_2 z t_1)
     (if (<= t_1 5e+66) (fma t_2 z (+ y x)) (+ (fma (- a 0.5) b y) x)))))

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+172) {
		tmp = fma(t_2, z, t_1);
	} else if (t_1 <= 5e+66) {
		tmp = fma(t_2, z, (y + x));
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	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+172)
		tmp = fma(t_2, z, t_1);
	elseif (t_1 <= 5e+66)
		tmp = fma(t_2, z, Float64(y + x));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	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+172], N[(t$95$2 * z + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 5e+66], N[(t$95$2 * z + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000001e172

   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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.4%

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

   if -5.0000000000000001e172 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999991e66

   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. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 4.99999999999999991e66 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, b \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+200}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -2e+200)
     t_2
     (if (<= t_1 5e+66) (fma (- 1.0 (log t)) z (+ y x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -2e+200) {
		tmp = t_2;
	} else if (t_1 <= 5e+66) {
		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 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -2e+200)
		tmp = t_2;
	elseif (t_1 <= 5e+66)
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+200], t$95$2, If[LessEqual[t$95$1, 5e+66], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.9999999999999999e200 or 4.99999999999999991e66 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 91.7

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

   5. Applied rewrites 91.7%

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

   if -1.9999999999999999e200 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999991e66

   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. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z + x\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- 1.0 (log t)) z) x)))
   (if (<= z -1.15e+193)
     t_1
     (if (<= z 1.1e+126) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((1.0 - log(t)) * z) + x;
	double tmp;
	if (z <= -1.15e+193) {
		tmp = t_1;
	} else if (z <= 1.1e+126) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(1.0 - log(t)) * z) + x)
	tmp = 0.0
	if (z <= -1.15e+193)
		tmp = t_1;
	elseif (z <= 1.1e+126)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.15e+193], t$95$1, If[LessEqual[z, 1.1e+126], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000007e193 or 1.09999999999999999e126 < 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 a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.5%

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

   if -1.15000000000000007e193 < z < 1.09999999999999999e126

   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. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

Alternative 9: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z y)))
   (if (<= z -1.3e+280)
     t_1
     (if (<= z 2.3e+86) (+ (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 <= -1.3e+280) {
		tmp = t_1;
	} else if (z <= 2.3e+86) {
		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 <= -1.3e+280)
		tmp = t_1;
	elseif (z <= 2.3e+86)
		tmp = Float64(fma(Float64(a - 0.5), b, 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, -1.3e+280], t$95$1, If[LessEqual[z, 2.3e+86], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e280 or 2.2999999999999999e86 < 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 a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 74.8%

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

   if -1.3e280 < z < 2.2999999999999999e86

   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. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 10: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -1.3e+280) t_1 (if (<= z 5e+186) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -1.3e+280) {
		tmp = t_1;
	} else if (z <= 5e+186) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -1.3e+280)
		tmp = t_1;
	elseif (z <= 5e+186)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.3e+280], t$95$1, If[LessEqual[z, 5e+186], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e280 or 4.99999999999999954e186 < z

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-log.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if -1.3e280 < z < 4.99999999999999954e186

   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. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 85.5

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

   5. Applied rewrites 85.5%

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

Alternative 11: 71.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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)))
   (if (<= t_1 -5e+172) t_2 (if (<= t_1 5e+191) (+ (fma -0.5 b 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);
	double tmp;
	if (t_1 <= -5e+172) {
		tmp = t_2;
	} else if (t_1 <= 5e+191) {
		tmp = fma(-0.5, b, 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, y)
	tmp = 0.0
	if (t_1 <= -5e+172)
		tmp = t_2;
	elseif (t_1 <= 5e+191)
		tmp = Float64(fma(-0.5, b, 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 + y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+172], t$95$2, If[LessEqual[t$95$1, 5e+191], N[(N[(-0.5 * b + 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) < -5.0000000000000001e172 or 5.0000000000000002e191 < (*.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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   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 85.1%

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

   if -5.0000000000000001e172 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000002e191

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot a + \left(y + x\right)\\ \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b a) (+ y x))))
   (if (<= (- a 0.5) -4e+16)
     t_1
     (if (<= (- a 0.5) 1000000000000.0) (+ (fma -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 = (b * a) + (y + x);
	double tmp;
	if ((a - 0.5) <= -4e+16) {
		tmp = t_1;
	} else if ((a - 0.5) <= 1000000000000.0) {
		tmp = fma(-0.5, b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * a) + Float64(y + x))
	tmp = 0.0
	if (Float64(a - 0.5) <= -4e+16)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 1000000000000.0)
		tmp = Float64(fma(-0.5, b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+16], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 1000000000000.0], N[(N[(-0.5 * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4e16 or 1e12 < (-.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 z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   if -4e16 < (-.f64 a #s(literal 1/2 binary64)) < 1e12

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;b \cdot a + \left(y + x\right)\\ \mathbf{elif}\;a - 0.5 \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a + \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.5000000000000007:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- a 0.5) -0.5000000000000007)
   (* b a)
   (if (<= (- a 0.5) 2e+45) (* -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) <= -0.5000000000000007) {
		tmp = b * a;
	} else if ((a - 0.5) <= 2e+45) {
		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) <= (-0.5000000000000007d0)) then
        tmp = b * a
    else if ((a - 0.5d0) <= 2d+45) 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) <= -0.5000000000000007) {
		tmp = b * a;
	} else if ((a - 0.5) <= 2e+45) {
		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) <= -0.5000000000000007:
		tmp = b * a
	elif (a - 0.5) <= 2e+45:
		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) <= -0.5000000000000007)
		tmp = Float64(b * a);
	elseif (Float64(a - 0.5) <= 2e+45)
		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) <= -0.5000000000000007)
		tmp = b * a;
	elseif ((a - 0.5) <= 2e+45)
		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], -0.5000000000000007], N[(b * a), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 2e+45], 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)) < -0.50000000000000067 or 1.9999999999999999e45 < (-.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. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   if -0.50000000000000067 < (-.f64 a #s(literal 1/2 binary64)) < 1.9999999999999999e45

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 22.5%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.5000000000000007:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{+49}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4.6e+49) (* b a) (if (<= a 3.5e+45) (fma -0.5 b y) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.6e+49) {
		tmp = b * a;
	} else if (a <= 3.5e+45) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4.6e+49)
		tmp = Float64(b * a);
	elseif (a <= 3.5e+45)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4.6e+49], N[(b * a), $MachinePrecision], If[LessEqual[a, 3.5e+45], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.60000000000000004e49 or 3.50000000000000023e45 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 48.6

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

   5. Applied rewrites 48.6%

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

   if -4.60000000000000004e49 < a < 3.50000000000000023e45

   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 53.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{+49}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-0.5 \cdot b + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) -4e+54) (+ (* -0.5 b) x) (fma (- a 0.5) b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= -4e+54) {
		tmp = (-0.5 * b) + x;
	} else {
		tmp = fma((a - 0.5), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y + x) <= -4e+54)
		tmp = Float64(Float64(-0.5 * b) + x);
	else
		tmp = fma(Float64(a - 0.5), b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], -4e+54], N[(N[(-0.5 * b), $MachinePrecision] + x), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.0000000000000003e54

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 35.2%

      \[\leadsto -0.5 \cdot b + x \]

   if -4.0000000000000003e54 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. 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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.9%

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

   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 58.0%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-0.5 \cdot b + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.7% accurate, 9.7× speedup

Math

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

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 Float64(fma(Float64(a - 0.5), b, y) + x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $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. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 77.1

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

5. Applied rewrites 77.1%

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

Alternative 17: 14.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

5. Applied rewrites 79.9%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 13.4%

   \[\leadsto -0.5 \cdot \color{blue}{b} \]
9. 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 2024244 
(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)))