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

Percentage Accurate: 99.8% → 99.8%

Time: 14.0s

Alternatives: 21

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

   7. log-lowering-log.f64 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
5. Add Preprocessing

Alternative 2: 39.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;z + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- a 0.5) b) (- (+ z (+ x y)) (* z (log t))))))
   (if (<= t_1 -2e+305)
     (* a b)
     (if (<= t_1 1e-21) (fma b -0.5 x) (if (<= t_1 2e+303) (+ z y) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a - 0.5) * b) + ((z + (x + y)) - (z * log(t)));
	double tmp;
	if (t_1 <= -2e+305) {
		tmp = a * b;
	} else if (t_1 <= 1e-21) {
		tmp = fma(b, -0.5, x);
	} else if (t_1 <= 2e+303) {
		tmp = z + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(a - 0.5) * b) + Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))))
	tmp = 0.0
	if (t_1 <= -2e+305)
		tmp = Float64(a * b);
	elseif (t_1 <= 1e-21)
		tmp = fma(b, -0.5, x);
	elseif (t_1 <= 2e+303)
		tmp = Float64(z + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+305], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(b * -0.5 + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(z + y), $MachinePrecision], N[(a * b), $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)) < -1.9999999999999999e305 or 2e303 < (+.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. *-commutative N/A

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

      2. *-lowering-*.f64 94.2

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

   5. Simplified 94.2%

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

   if -1.9999999999999999e305 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 9.99999999999999908e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 56.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2}} + x \]

      3. accelerator-lowering-fma.f64 43.1

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

   8. Simplified 43.1%

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

   if 9.99999999999999908e-22 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2e303

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 90.2

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

   7. Simplified 90.2%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 39.4%

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

   11. Taylor expanded in b around 0

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

       1. +-lowering-+.f64 28.1

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

   13. Simplified 28.1%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;z + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- a 0.5) b) (- (+ z (+ x y)) (* z (log t))))))
   (if (<= t_1 -2e+305)
     (* a b)
     (if (<= t_1 20.0) (fma b -0.5 x) (fma b a y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a - 0.5) * b) + ((z + (x + y)) - (z * log(t)));
	double tmp;
	if (t_1 <= -2e+305) {
		tmp = a * b;
	} else if (t_1 <= 20.0) {
		tmp = fma(b, -0.5, x);
	} else {
		tmp = fma(b, a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(a - 0.5) * b) + Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))))
	tmp = 0.0
	if (t_1 <= -2e+305)
		tmp = Float64(a * b);
	elseif (t_1 <= 20.0)
		tmp = fma(b, -0.5, x);
	else
		tmp = fma(b, a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+305], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(b * -0.5 + x), $MachinePrecision], N[(b * a + y), $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)) < -1.9999999999999999e305

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.9999999999999999e305 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 20

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 55.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2}} + x \]

      3. accelerator-lowering-fma.f64 42.1

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

   8. Simplified 42.1%

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

   if 20 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 90.7

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

   7. Simplified 90.7%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 47.7%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 46.8

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

   13. Simplified 46.8%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq 20:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+111}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+88}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma z (- 1.0 (log t)) y)))
   (if (<= t_1 -6e+111)
     (+ y (fma b (+ a -0.5) x))
     (if (<= t_1 5e+88) (+ x (fma b -0.5 t_2)) (fma b (+ a -0.5) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma(z, (1.0 - log(t)), y);
	double tmp;
	if (t_1 <= -6e+111) {
		tmp = y + fma(b, (a + -0.5), x);
	} else if (t_1 <= 5e+88) {
		tmp = x + fma(b, -0.5, t_2);
	} else {
		tmp = fma(b, (a + -0.5), t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(z, Float64(1.0 - log(t)), y)
	tmp = 0.0
	if (t_1 <= -6e+111)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	elseif (t_1 <= 5e+88)
		tmp = Float64(x + fma(b, -0.5, t_2));
	else
		tmp = fma(b, Float64(a + -0.5), t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+111], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+88], N[(x + N[(b * -0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -6e111

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 94.4

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

   5. Simplified 94.4%

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

   if -6e111 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.99999999999999997e88

   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. +-lowering-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. log-rec N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

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

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Simplified 96.3%

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

   if 4.99999999999999997e88 < (*.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

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

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

      8. log-rec N/A

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

      9. distribute-rgt-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. log-rec N/A

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

   7. Simplified 90.7%

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

Alternative 5: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -6e+111)
     t_2
     (if (<= t_1 2e+141) (fma z (- 1.0 (log t)) (+ x y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = y + fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -6e+111) {
		tmp = t_2;
	} else if (t_1 <= 2e+141) {
		tmp = fma(z, (1.0 - log(t)), (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(y + fma(b, Float64(a + -0.5), x))
	tmp = 0.0
	if (t_1 <= -6e+111)
		tmp = t_2;
	elseif (t_1 <= 2e+141)
		tmp = fma(z, Float64(1.0 - log(t)), Float64(x + y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+111], t$95$2, If[LessEqual[t$95$1, 2e+141], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -6e111 or 2.00000000000000003e141 < (*.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. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 92.2

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

   5. Simplified 92.2%

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

   if -6e111 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.00000000000000003e141

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -6 \cdot 10^{+111}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;a - 0.5 \leq -5000000000000:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;a - 0.5 \leq 10^{+38}:\\ \;\;\;\;x + \mathsf{fma}\left(b, -0.5, \mathsf{fma}\left(z, t\_1, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(a, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (- a 0.5) -5000000000000.0)
     (+ y (fma b (+ a -0.5) x))
     (if (<= (- a 0.5) 1e+38)
       (+ x (fma b -0.5 (fma z t_1 y)))
       (fma z t_1 (fma a b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((a - 0.5) <= -5000000000000.0) {
		tmp = y + fma(b, (a + -0.5), x);
	} else if ((a - 0.5) <= 1e+38) {
		tmp = x + fma(b, -0.5, fma(z, t_1, y));
	} else {
		tmp = fma(z, t_1, fma(a, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -5000000000000.0)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	elseif (Float64(a - 0.5) <= 1e+38)
		tmp = Float64(x + fma(b, -0.5, fma(z, t_1, y)));
	else
		tmp = fma(z, t_1, fma(a, b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5000000000000.0], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+38], N[(x + N[(b * -0.5 + N[(z * t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + N[(a * b + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

   if -5e12 < (-.f64 a #s(literal 1/2 binary64)) < 9.99999999999999977e37

   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. +-lowering-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. log-rec N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

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

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

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

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

      17. associate-+l+ N/A

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

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

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

   5. Simplified 98.3%

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

   if 9.99999999999999977e37 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r- N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

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

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

      14. --lowering--.f64 N/A

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

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

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

      16. +-commutative N/A

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

      17. accelerator-lowering-fma.f64 84.6

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

   10. Simplified 84.6%

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

Alternative 7: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(z + x\right)\\ \mathbf{elif}\;x + y \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, b \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(a, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -1e+107)
     (+ (* (- a 0.5) b) (+ z x))
     (if (<= (+ x y) 1e+16)
       (fma z t_1 (* b (+ a -0.5)))
       (fma z t_1 (fma a b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((x + y) <= -1e+107) {
		tmp = ((a - 0.5) * b) + (z + x);
	} else if ((x + y) <= 1e+16) {
		tmp = fma(z, t_1, (b * (a + -0.5)));
	} else {
		tmp = fma(z, t_1, fma(a, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(x + y) <= -1e+107)
		tmp = Float64(Float64(Float64(a - 0.5) * b) + Float64(z + x));
	elseif (Float64(x + y) <= 1e+16)
		tmp = fma(z, t_1, Float64(b * Float64(a + -0.5)));
	else
		tmp = fma(z, t_1, fma(a, b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -1e+107], N[(N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+16], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + N[(a * b + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -9.9999999999999997e106

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 68.2%

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

   if -9.9999999999999997e106 < (+.f64 x y) < 1e16

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

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

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

      8. log-rec N/A

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

      9. distribute-rgt-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. log-rec N/A

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

   7. Simplified 92.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. --lowering--.f64 N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 87.2

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

   10. Simplified 87.2%

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

   if 1e16 < (+.f64 x y)

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 94.8

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

   7. Simplified 94.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r- N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

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

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

      14. --lowering--.f64 N/A

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

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

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

      16. +-commutative N/A

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

      17. accelerator-lowering-fma.f64 62.8

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

   10. Simplified 62.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(z + x\right)\\ \mathbf{elif}\;x + y \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, b \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 21.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq -2 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* (- a 0.5) b) (- (+ z (+ x y)) (* z (log t)))) -2e-151) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((a - 0.5) * b) + ((z + (x + y)) - (z * log(t)))) <= -2e-151) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((a - 0.5d0) * b) + ((z + (x + y)) - (z * log(t)))) <= (-2d-151)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((a - 0.5) * b) + ((z + (x + y)) - (z * Math.log(t)))) <= -2e-151) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (((a - 0.5) * b) + ((z + (x + y)) - (z * math.log(t)))) <= -2e-151:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a - 0.5) * b) + Float64(Float64(z + Float64(x + y)) - Float64(z * log(t)))) <= -2e-151)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((((a - 0.5) * b) + ((z + (x + y)) - (z * log(t)))) <= -2e-151)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision] + N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-151], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.9999999999999999e-151

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 24.3%

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

   if -1.9999999999999999e-151 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 22.6%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b + \left(\left(z + \left(x + y\right)\right) - z \cdot \log t\right) \leq -2 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - z \cdot \log t \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ x y)) (* z (log t))) -2e-151)
   (fma (+ a -0.5) b x)
   (fma b (+ a -0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z + (x + y)) - (z * log(t))) <= -2e-151) {
		tmp = fma((a + -0.5), b, x);
	} else {
		tmp = fma(b, (a + -0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(z + Float64(x + y)) - Float64(z * log(t))) <= -2e-151)
		tmp = fma(Float64(a + -0.5), b, x);
	else
		tmp = fma(b, Float64(a + -0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-151], N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -1.9999999999999999e-151

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.9%

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 62.9

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

   7. Applied egg-rr 62.9%

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

   if -1.9999999999999999e-151 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 52.6

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

   8. Simplified 52.6%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(x + y\right)\right) - z \cdot \log t \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(b, a + -0.5, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, \mathsf{fma}\left(z, t\_1, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -2e-151)
     (fma z t_1 (fma b (+ a -0.5) x))
     (fma b (+ a -0.5) (fma z t_1 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((x + y) <= -2e-151) {
		tmp = fma(z, t_1, fma(b, (a + -0.5), x));
	} else {
		tmp = fma(b, (a + -0.5), fma(z, t_1, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(x + y) <= -2e-151)
		tmp = fma(z, t_1, fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(b, Float64(a + -0.5), fma(z, t_1, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -2e-151], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + N[(z * t$95$1 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.9999999999999999e-151

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

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

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

   5. Simplified 82.1%

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

   if -1.9999999999999999e-151 < (+.f64 x y)

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

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

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

      8. log-rec N/A

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

      9. distribute-rgt-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. log-rec N/A

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

   7. Simplified 77.8%

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

Alternative 11: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, y\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+159}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -3.2e+158)
     (fma z t_1 y)
     (if (<= z 8.8e+159) (+ y (fma b (+ a -0.5) x)) (fma z t_1 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -3.2e+158) {
		tmp = fma(z, t_1, y);
	} else if (z <= 8.8e+159) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -3.2e+158)
		tmp = fma(z, t_1, y);
	elseif (z <= 8.8e+159)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+158], N[(z * t$95$1 + y), $MachinePrecision], If[LessEqual[z, 8.8e+159], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999995e158

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 79.2

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

   5. Simplified 79.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 68.9

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

   8. Simplified 68.9%

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

   if -3.19999999999999995e158 < z < 8.7999999999999997e159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 90.2

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

   5. Simplified 90.2%

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

   if 8.7999999999999997e159 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 80.4

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

   8. Simplified 80.4%

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

Alternative 12: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, x\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+159}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) x)))
   (if (<= z -3.7e+169)
     t_1
     (if (<= z 7e+159) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), x);
	double tmp;
	if (z <= -3.7e+169) {
		tmp = t_1;
	} else if (z <= 7e+159) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), x)
	tmp = 0.0
	if (z <= -3.7e+169)
		tmp = t_1;
	elseif (z <= 7e+159)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -3.7e+169], t$95$1, If[LessEqual[z, 7e+159], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.70000000000000001e169 or 6.9999999999999999e159 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 71.8

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

   8. Simplified 71.8%

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

   if -3.70000000000000001e169 < z < 6.9999999999999999e159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 89.9

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

   5. Simplified 89.9%

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

Alternative 13: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+247}:\\ \;\;\;\;\mathsf{fma}\left(z, -\log t, z\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+181}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.65e+247)
   (fma z (- (log t)) z)
   (if (<= z 1.8e+181) (+ y (fma b (+ a -0.5) x)) (- z (* z (log t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.65e+247) {
		tmp = fma(z, -log(t), z);
	} else if (z <= 1.8e+181) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = z - (z * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.65e+247)
		tmp = fma(z, Float64(-log(t)), z);
	elseif (z <= 1.8e+181)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = Float64(z - Float64(z * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.65e+247], N[(z * (-N[Log[t], $MachinePrecision]) + z), $MachinePrecision], If[LessEqual[z, 1.8e+181], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.65000000000000001e247

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

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

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

      8. log-rec N/A

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

      9. distribute-rgt-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. log-rec N/A

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. log-rec N/A

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

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

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

      9. log-lowering-log.f64 74.9

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

   10. Simplified 74.9%

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

   if -1.65000000000000001e247 < z < 1.79999999999999992e181

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 86.6

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

   5. Simplified 86.6%

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

   if 1.79999999999999992e181 < z

   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 inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

      14. log-lowering-log.f64 76.2

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

   5. Simplified 76.2%

      \[\leadsto \color{blue}{z - z \cdot \log t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+181}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- z (* z (log t)))))
   (if (<= z -1.75e+247)
     t_1
     (if (<= z 5.2e+181) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z - (z * log(t));
	double tmp;
	if (z <= -1.75e+247) {
		tmp = t_1;
	} else if (z <= 5.2e+181) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z - Float64(z * log(t)))
	tmp = 0.0
	if (z <= -1.75e+247)
		tmp = t_1;
	elseif (z <= 5.2e+181)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e+247], t$95$1, If[LessEqual[z, 5.2e+181], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7500000000000001e247 or 5.2e181 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

      14. log-lowering-log.f64 75.6

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

   5. Simplified 75.6%

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

   if -1.7500000000000001e247 < z < 5.2e181

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 86.6

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

   5. Simplified 86.6%

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

Alternative 15: 57.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+234}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+209}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -4e+234) (* a b) (if (<= t_1 2e+209) (+ x y) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -4e+234) {
		tmp = a * b;
	} else if (t_1 <= 2e+209) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-4d+234)) then
        tmp = a * b
    else if (t_1 <= 2d+209) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -4e+234) {
		tmp = a * b;
	} else if (t_1 <= 2e+209) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -4e+234:
		tmp = a * b
	elif t_1 <= 2e+209:
		tmp = x + y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -4e+234)
		tmp = Float64(a * b);
	elseif (t_1 <= 2e+209)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -4e+234)
		tmp = a * b;
	elseif (t_1 <= 2e+209)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+234], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 2e+209], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.00000000000000007e234 or 2.0000000000000001e209 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   if -4.00000000000000007e234 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.0000000000000001e209

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 81.8

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

   5. Simplified 81.8%

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

   6. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 54.0

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

   8. Simplified 54.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -4 \cdot 10^{+234}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+209}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -6 \cdot 10^{-33}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 10^{+16}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -6e-33)
   (+ x (* a b))
   (if (<= (+ x y) 1e+16) (* b (+ a -0.5)) (fma b a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -6e-33) {
		tmp = x + (a * b);
	} else if ((x + y) <= 1e+16) {
		tmp = b * (a + -0.5);
	} else {
		tmp = fma(b, a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -6e-33)
		tmp = Float64(x + Float64(a * b));
	elseif (Float64(x + y) <= 1e+16)
		tmp = Float64(b * Float64(a + -0.5));
	else
		tmp = fma(b, a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -6e-33], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+16], N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision], N[(b * a + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -6.0000000000000003e-33

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   if -6.0000000000000003e-33 < (+.f64 x y) < 1e16

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 58.7

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

   5. Simplified 58.7%

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

   if 1e16 < (+.f64 x y)

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 94.8

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

   7. Simplified 94.8%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 48.0%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 47.3

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

   13. Simplified 47.3%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -6 \cdot 10^{-33}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 10^{+16}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;b \leq -2.75 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (+ a -0.5))))
   (if (<= b -2.75e+150) t_1 (if (<= b 1.8e+163) (fma a b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a + -0.5);
	double tmp;
	if (b <= -2.75e+150) {
		tmp = t_1;
	} else if (b <= 1.8e+163) {
		tmp = fma(a, b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (b <= -2.75e+150)
		tmp = t_1;
	elseif (b <= 1.8e+163)
		tmp = fma(a, b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.75e+150], t$95$1, If[LessEqual[b, 1.8e+163], N[(a * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.75000000000000008e150 or 1.79999999999999989e163 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 83.7

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

   5. Simplified 83.7%

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

   if -2.75000000000000008e150 < b < 1.79999999999999989e163

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      7. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 95.2

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

   7. Simplified 95.2%

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

   8. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

      4. +-lowering-+.f64 68.8

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

   10. Simplified 68.8%

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

Alternative 18: 65.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a + -0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -7.6e-33) (fma a b (+ x y)) (fma b (+ a -0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -7.6e-33) {
		tmp = fma(a, b, (x + y));
	} else {
		tmp = fma(b, (a + -0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -7.6e-33)
		tmp = fma(a, b, Float64(x + y));
	else
		tmp = fma(b, Float64(a + -0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -7.6e-33], N[(a * b + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999988e-33

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(z + \left(\color{blue}{\left(x + y\right)} - z \cdot \log t\right)\right) + \left(a - \frac{1}{2}\right) \cdot b \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(z + \left(\left(x + y\right) - \color{blue}{z \cdot \log t}\right)\right) + \left(a - \frac{1}{2}\right) \cdot b \]

      7. log-lowering-log.f64 100.0

         \[\leadsto \left(z + \left(\left(x + y\right) - z \cdot \color{blue}{\log t}\right)\right) + \left(a - 0.5\right) \cdot b \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]

   5. Taylor expanded in a around inf

      \[\leadsto \left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \color{blue}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \color{blue}{b \cdot a} \]

      2. *-lowering-*.f64 90.8

         \[\leadsto \left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \color{blue}{b \cdot a} \]

   7. Simplified 90.8%

      \[\leadsto \left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \color{blue}{b \cdot a} \]

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + a \cdot b\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + a \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot b + \left(x + y\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x + y\right)} \]

      4. +-lowering-+.f64 77.7

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x + y}\right) \]

   10. Simplified 77.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x + y\right)} \]

   if -7.59999999999999988e-33 < x

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{y + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{y + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto y + \color{blue}{\mathsf{fma}\left(b, a - \frac{1}{2}, x\right)} \]

      5. sub-neg N/A

         \[\leadsto y + \mathsf{fma}\left(b, \color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x\right) \]

      6. metadata-eval N/A

         \[\leadsto y + \mathsf{fma}\left(b, a + \color{blue}{\frac{-1}{2}}, x\right) \]

      7. +-lowering-+.f64 74.3

         \[\leadsto y + \mathsf{fma}\left(b, \color{blue}{a + -0.5}, x\right) \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{y + \mathsf{fma}\left(b, a + -0.5, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + b \cdot \left(a - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, a - \frac{1}{2}, y\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, y\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, a + \color{blue}{\frac{-1}{2}}, y\right) \]

      5. +-lowering-+.f64 57.7

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{a + -0.5}, y\right) \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a + -0.5, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ y + \mathsf{fma}\left(b, a + -0.5, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ y (fma b (+ a -0.5) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return y + fma(b, (a + -0.5), x);
}

Julia

function code(x, y, z, t, a, b)
	return Float64(y + fma(b, Float64(a + -0.5), x))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{y + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{y + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto y + \color{blue}{\mathsf{fma}\left(b, a - \frac{1}{2}, x\right)} \]

   5. sub-neg N/A

      \[\leadsto y + \mathsf{fma}\left(b, \color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x\right) \]

   6. metadata-eval N/A

      \[\leadsto y + \mathsf{fma}\left(b, a + \color{blue}{\frac{-1}{2}}, x\right) \]

   7. +-lowering-+.f64 77.7

      \[\leadsto y + \mathsf{fma}\left(b, \color{blue}{a + -0.5}, x\right) \]

5. Simplified 77.7%

   \[\leadsto \color{blue}{y + \mathsf{fma}\left(b, a + -0.5, x\right)} \]
6. Add Preprocessing

Alternative 20: 41.9% accurate, 31.5× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + y;
}

Python

def code(x, y, z, t, a, b):
	return x + y

Julia

function code(x, y, z, t, a, b)
	return Float64(x + y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(x + y\right) + z\right)} + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t\right)} \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \left(x + y\right) + \color{blue}{\left(z - z \cdot \log t\right)} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(x + y\right) + \left(\color{blue}{z \cdot 1} - z \cdot \log t\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \left(x + y\right) + \color{blue}{z \cdot \left(1 - \log t\right)} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right) + \left(x + y\right)} \]

   8. sub-neg N/A

      \[\leadsto z \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\log t\right)\right)\right)} + \left(x + y\right) \]

   9. mul-1-neg N/A

      \[\leadsto z \cdot \left(1 + \color{blue}{-1 \cdot \log t}\right) + \left(x + y\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1 + -1 \cdot \log t, x + y\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(z, 1 + \color{blue}{\left(\mathsf{neg}\left(\log t\right)\right)}, x + y\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{1 - \log t}, x + y\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{1 - \log t}, x + y\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, 1 - \color{blue}{\log t}, x + y\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, 1 - \log t, \color{blue}{y + x}\right) \]

   16. +-lowering-+.f64 64.7

       \[\leadsto \mathsf{fma}\left(z, 1 - \log t, \color{blue}{y + x}\right) \]

5. Simplified 64.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1 - \log t, y + x\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + y} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 43.0

      \[\leadsto \color{blue}{x + y} \]

8. Simplified 43.0%

   \[\leadsto \color{blue}{x + y} \]
9. Add Preprocessing

Alternative 21: 21.5% accurate, 126.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 24.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + y) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + log(t)))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(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)))