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

Percentage Accurate: 99.9% → 99.9%

Time: 12.0s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := 1 - \log t\\ t_3 := \mathsf{fma}\left(t\_2, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+93}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, x\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b))
        (t_2 (- 1.0 (log t)))
        (t_3 (fma t_2 z (fma (- a 0.5) b y))))
   (if (<= t_1 -2e+93) t_3 (if (<= t_1 2e+72) (+ (fma t_2 z x) y) t_3))))

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 = 1.0 - log(t);
	double t_3 = fma(t_2, z, fma((a - 0.5), b, y));
	double tmp;
	if (t_1 <= -2e+93) {
		tmp = t_3;
	} else if (t_1 <= 2e+72) {
		tmp = fma(t_2, z, x) + y;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(1.0 - log(t))
	t_3 = fma(t_2, z, fma(Float64(a - 0.5), b, y))
	tmp = 0.0
	if (t_1 <= -2e+93)
		tmp = t_3;
	elseif (t_1 <= 2e+72)
		tmp = Float64(fma(t_2, z, x) + y);
	else
		tmp = t_3;
	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[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+93], t$95$3, If[LessEqual[t$95$1, 2e+72], N[(N[(t$95$2 * z + x), $MachinePrecision] + y), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000009e93 or 1.99999999999999989e72 < (*.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 x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.1%

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

   if -2.00000000000000009e93 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999989e72

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

Alternative 3: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right) + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1e+95)
     (+ (fma (- a 0.5) b y) x)
     (if (<= t_1 5e+79)
       (+ (fma (- 1.0 (log t)) z x) y)
       (fma b a (fma -0.5 b (+ y x)))))))

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 <= -1e+95) {
		tmp = fma((a - 0.5), b, y) + x;
	} else if (t_1 <= 5e+79) {
		tmp = fma((1.0 - log(t)), z, x) + y;
	} else {
		tmp = fma(b, a, fma(-0.5, b, (y + x)));
	}
	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 <= -1e+95)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	elseif (t_1 <= 5e+79)
		tmp = Float64(fma(Float64(1.0 - log(t)), z, x) + y);
	else
		tmp = fma(b, a, fma(-0.5, b, Float64(y + x)));
	end
	return 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, -1e+95], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+79], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision] + y), $MachinePrecision], N[(b * a + N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.00000000000000002e95

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.00000000000000002e95 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5e79

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   if 5e79 < (*.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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 85.5

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

   7. Applied rewrites 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 85.5

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

   9. Applied rewrites 85.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right) + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) -5e-78)
   (fma (- a 0.5) b (fma (- z) (log t) (+ z x)))
   (fma (- 1.0 (log t)) z (fma (- a 0.5) b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= -5e-78) {
		tmp = fma((a - 0.5), b, fma(-z, log(t), (z + x)));
	} else {
		tmp = fma((1.0 - log(t)), z, fma((a - 0.5), b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y + x) <= -5e-78)
		tmp = fma(Float64(a - 0.5), b, fma(Float64(-z), log(t), Float64(z + x)));
	else
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], -5e-78], N[(N[(a - 0.5), $MachinePrecision] * b + N[((-z) * N[Log[t], $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999996e-78

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 68.9

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

   7. Applied rewrites 68.9%

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

   if -4.9999999999999996e-78 < (+.f64 x y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.9

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.9

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z y)))
   (if (<= z -1.75e+222)
     t_1
     (if (<= z 1.25e+200) (fma b a (fma -0.5 b (+ y x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, y);
	double tmp;
	if (z <= -1.75e+222) {
		tmp = t_1;
	} else if (z <= 1.25e+200) {
		tmp = fma(b, a, fma(-0.5, b, (y + x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, y)
	tmp = 0.0
	if (z <= -1.75e+222)
		tmp = t_1;
	elseif (z <= 1.25e+200)
		tmp = fma(b, a, fma(-0.5, b, Float64(y + x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision]}, If[LessEqual[z, -1.75e+222], t$95$1, If[LessEqual[z, 1.25e+200], N[(b * a + N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7499999999999999e222 or 1.25000000000000005e200 < 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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.1%

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

   if -1.7499999999999999e222 < z < 1.25000000000000005e200

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.1

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

   7. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 88.1

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

   9. Applied rewrites 88.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{elif}\;z \leq 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5e+224)
   (- z (* (log t) z))
   (if (<= z 1e+200)
     (fma b a (fma -0.5 b (+ y x)))
     (fma (- 1.0 (log t)) z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5e+224) {
		tmp = z - (log(t) * z);
	} else if (z <= 1e+200) {
		tmp = fma(b, a, fma(-0.5, b, (y + x)));
	} else {
		tmp = fma((1.0 - log(t)), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.5e+224)
		tmp = Float64(z - Float64(log(t) * z));
	elseif (z <= 1e+200)
		tmp = fma(b, a, fma(-0.5, b, Float64(y + x)));
	else
		tmp = fma(Float64(1.0 - log(t)), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.5e+224], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+200], N[(b * a + N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e224

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.5000000000000001e224 < z < 9.9999999999999997e199

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.1

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

   7. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 88.1

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

   9. Applied rewrites 88.1%

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

   if 9.9999999999999997e199 < 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 b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{elif}\;z \leq 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5e+224) (- z (* (log t) z)) (fma b a (fma -0.5 b (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5e+224) {
		tmp = z - (log(t) * z);
	} else {
		tmp = fma(b, a, fma(-0.5, b, (y + x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.5e+224)
		tmp = Float64(z - Float64(log(t) * z));
	else
		tmp = fma(b, a, fma(-0.5, b, Float64(y + x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.5e+224], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5000000000000001e224

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.5000000000000001e224 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.2

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

   7. Applied rewrites 84.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 84.3

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

   9. Applied rewrites 84.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(-0.5, b, y + x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+178}:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -2e+247) t_1 (if (<= t_1 1e+178) (+ (* -0.5 b) (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+247) {
		tmp = t_1;
	} else if (t_1 <= 1e+178) {
		tmp = (-0.5 * b) + (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-2d+247)) then
        tmp = t_1
    else if (t_1 <= 1d+178) then
        tmp = ((-0.5d0) * b) + (y + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+247) {
		tmp = t_1;
	} else if (t_1 <= 1e+178) {
		tmp = (-0.5 * b) + (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -2e+247:
		tmp = t_1
	elif t_1 <= 1e+178:
		tmp = (-0.5 * b) + (y + x)
	else:
		tmp = t_1
	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 <= -2e+247)
		tmp = t_1;
	elseif (t_1 <= 1e+178)
		tmp = Float64(Float64(-0.5 * b) + Float64(y + x));
	else
		tmp = t_1;
	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 <= -2e+247)
		tmp = t_1;
	elseif (t_1 <= 1e+178)
		tmp = (-0.5 * b) + (y + x);
	else
		tmp = t_1;
	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, -2e+247], t$95$1, If[LessEqual[t$95$1, 1e+178], N[(N[(-0.5 * b), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.9999999999999999e247 or 1.0000000000000001e178 < (*.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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if -1.9999999999999999e247 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.0000000000000001e178

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 65.2

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

   10. Applied rewrites 65.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+247}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+178}:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+178}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -5e+154) t_1 (if (<= t_1 1e+178) (+ y x) t_1))))

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 <= -5e+154) {
		tmp = t_1;
	} else if (t_1 <= 1e+178) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-5d+154)) then
        tmp = t_1
    else if (t_1 <= 1d+178) then
        tmp = y + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -5e+154) {
		tmp = t_1;
	} else if (t_1 <= 1e+178) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -5e+154:
		tmp = t_1
	elif t_1 <= 1e+178:
		tmp = y + x
	else:
		tmp = t_1
	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 <= -5e+154)
		tmp = t_1;
	elseif (t_1 <= 1e+178)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -5e+154)
		tmp = t_1;
	elseif (t_1 <= 1e+178)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+154], t$95$1, If[LessEqual[t$95$1, 1e+178], N[(y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.00000000000000004e154 or 1.0000000000000001e178 < (*.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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -5.00000000000000004e154 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.0000000000000001e178

   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. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.4%

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

Alternative 12: 57.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+247}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+178}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -2e+247) (* a b) (if (<= t_1 1e+178) (+ y x) (* 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 <= -2e+247) {
		tmp = a * b;
	} else if (t_1 <= 1e+178) {
		tmp = y + x;
	} 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 <= (-2d+247)) then
        tmp = a * b
    else if (t_1 <= 1d+178) then
        tmp = y + x
    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 <= -2e+247) {
		tmp = a * b;
	} else if (t_1 <= 1e+178) {
		tmp = y + x;
	} 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 <= -2e+247:
		tmp = a * b
	elif t_1 <= 1e+178:
		tmp = y + x
	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 <= -2e+247)
		tmp = Float64(a * b);
	elseif (t_1 <= 1e+178)
		tmp = Float64(y + x);
	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 <= -2e+247)
		tmp = a * b;
	elseif (t_1 <= 1e+178)
		tmp = y + x;
	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, -2e+247], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+178], N[(y + x), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.9999999999999999e247 or 1.0000000000000001e178 < (*.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 a around inf

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

      1. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.9999999999999999e247 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.0000000000000001e178

   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. *-commutative N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 61.6%

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

Alternative 13: 77.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + \left(y + x\right)\\ \mathbf{if}\;a - 0.5 \leq -10000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq -0.4999995:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (+ y x))))
   (if (<= (- a 0.5) -10000000000000.0)
     t_1
     (if (<= (- a 0.5) -0.4999995) (+ (* -0.5 b) (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (y + x);
	double tmp;
	if ((a - 0.5) <= -10000000000000.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4999995) {
		tmp = (-0.5 * b) + (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (y + x)
    if ((a - 0.5d0) <= (-10000000000000.0d0)) then
        tmp = t_1
    else if ((a - 0.5d0) <= (-0.4999995d0)) then
        tmp = ((-0.5d0) * b) + (y + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (y + x);
	double tmp;
	if ((a - 0.5) <= -10000000000000.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4999995) {
		tmp = (-0.5 * b) + (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (y + x)
	tmp = 0
	if (a - 0.5) <= -10000000000000.0:
		tmp = t_1
	elif (a - 0.5) <= -0.4999995:
		tmp = (-0.5 * b) + (y + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(y + x))
	tmp = 0.0
	if (Float64(a - 0.5) <= -10000000000000.0)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.4999995)
		tmp = Float64(Float64(-0.5 * b) + Float64(y + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (y + x);
	tmp = 0.0;
	if ((a - 0.5) <= -10000000000000.0)
		tmp = t_1;
	elseif ((a - 0.5) <= -0.4999995)
		tmp = (-0.5 * b) + (y + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -10000000000000.0], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4999995], N[(N[(-0.5 * b), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e13 or -0.499999499999999986 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 83.3

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

   10. Applied rewrites 83.3%

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

   if -1e13 < (-.f64 a #s(literal 1/2 binary64)) < -0.499999499999999986

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 72.7

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 72.5

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

   10. Applied rewrites 72.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -10000000000000:\\ \;\;\;\;a \cdot b + \left(y + x\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.4999995:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.3% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 78.2

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

7. Applied rewrites 78.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 78.3

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

9. Applied rewrites 78.3%

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

10. Final simplification 78.3%

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

Alternative 15: 78.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 78.2

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

5. Applied rewrites 78.2%

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

Alternative 16: 41.3% accurate, 31.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in b around 0

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

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

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

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

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

   5. *-rgt-identity N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 46.1%

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

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + y) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + log(t)))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024235 
(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)))