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

Percentage Accurate: 99.8% → 99.9%

Time: 15.9s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ y (fma z (- 1.0 (log t)) (fma b (+ a -0.5) x))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. associate--l+ N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. log-rec N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. associate-+r+ N/A

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

5. Simplified 99.9%

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

Alternative 2: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ t_2 := \left(z + \left(y + x\right)\right) - z \cdot \log t\\ \mathbf{if}\;t\_2 \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(b, a + -0.5, x\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+120}:\\ \;\;\;\;y + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(z, t\_1, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))) (t_2 (- (+ z (+ y x)) (* z (log t)))))
   (if (<= t_2 1e-37)
     (fma z t_1 (fma b (+ a -0.5) x))
     (if (<= t_2 2e+120) (+ y (* b (- a 0.5))) (fma b a (fma z t_1 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double t_2 = (z + (y + x)) - (z * log(t));
	double tmp;
	if (t_2 <= 1e-37) {
		tmp = fma(z, t_1, fma(b, (a + -0.5), x));
	} else if (t_2 <= 2e+120) {
		tmp = y + (b * (a - 0.5));
	} else {
		tmp = fma(b, a, fma(z, t_1, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	t_2 = Float64(Float64(z + Float64(y + x)) - Float64(z * log(t)))
	tmp = 0.0
	if (t_2 <= 1e-37)
		tmp = fma(z, t_1, fma(b, Float64(a + -0.5), x));
	elseif (t_2 <= 2e+120)
		tmp = Float64(y + Float64(b * Float64(a - 0.5)));
	else
		tmp = fma(b, a, fma(z, t_1, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-37], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+120], N[(y + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(z * t$95$1 + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < 1.00000000000000007e-37

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

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

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

   5. Simplified 79.4%

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

   if 1.00000000000000007e-37 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < 2e120

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 79.7%

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

   if 2e120 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r- N/A

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

      4. *-commutative N/A

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

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

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

      6. log-rec N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. log-rec N/A

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

      12. sub-neg N/A

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

      13. associate-+l+ N/A

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

      14. *-commutative N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. --lowering--.f64 N/A

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

      19. log-lowering-log.f64 77.3

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

   8. Simplified 77.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, x\right)\right)\\ \mathbf{elif}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq 2 \cdot 10^{+120}:\\ \;\;\;\;y + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(z, 1 - \log t, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (+ y (fma b (+ a -0.5) x))))
   (if (<= t_1 -4e-7)
     t_2
     (if (<= t_1 5e+162) (fma z (- 1.0 (log t)) (+ y x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = y + fma(b, (a + -0.5), x);
	double tmp;
	if (t_1 <= -4e-7) {
		tmp = t_2;
	} else if (t_1 <= 5e+162) {
		tmp = fma(z, (1.0 - log(t)), (y + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(y + fma(b, Float64(a + -0.5), x))
	tmp = 0.0
	if (t_1 <= -4e-7)
		tmp = t_2;
	elseif (t_1 <= 5e+162)
		tmp = fma(z, Float64(1.0 - log(t)), Float64(y + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-7], t$95$2, If[LessEqual[t$95$1, 5e+162], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -3.9999999999999998e-7 or 4.9999999999999997e162 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 90.6

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

   5. Simplified 90.6%

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

   if -3.9999999999999998e-7 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999997e162

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 95.5

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

   5. Simplified 95.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -4 \cdot 10^{-7}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq -4 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot \left(a - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ z (+ y x)) (* z (log t))) -4e-113)
   (fma (+ a -0.5) b x)
   (+ y (* b (- a 0.5)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z + (y + x)) - (z * log(t))) <= -4e-113) {
		tmp = fma((a + -0.5), b, x);
	} else {
		tmp = y + (b * (a - 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(z + Float64(y + x)) - Float64(z * log(t))) <= -4e-113)
		tmp = fma(Float64(a + -0.5), b, x);
	else
		tmp = Float64(y + Float64(b * Float64(a - 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-113], N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(y + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -3.99999999999999991e-113

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 54.1%

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 54.1

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

   7. Applied egg-rr 54.1%

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

   if -3.99999999999999991e-113 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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 y around inf

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

   5. Simplified 56.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + \left(y + x\right)\right) - z \cdot \log t \leq -4 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot \left(a - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 22.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) + b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ z (+ y x)) (* z (log t))) (* b (- a 0.5))) -5e-108) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((z + (y + x)) - (z * log(t))) + (b * (a - 0.5))) <= -5e-108) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((z + (y + x)) - (z * log(t))) + (b * (a - 0.5d0))) <= (-5d-108)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((z + (y + x)) - (z * Math.log(t))) + (b * (a - 0.5))) <= -5e-108) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (((z + (y + x)) - (z * math.log(t))) + (b * (a - 0.5))) <= -5e-108:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(y + x)) - Float64(z * log(t))) + Float64(b * Float64(a - 0.5))) <= -5e-108)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((((z + (y + x)) - (z * log(t))) + (b * (a - 0.5))) <= -5e-108)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-108], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -5e-108

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 23.6%

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

   if -5e-108 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 24.9%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) + b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;y + x \leq -4 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(b, a + -0.5, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ y x) -4e-113)
     (fma z t_1 (fma b (+ a -0.5) x))
     (fma z t_1 (fma b (+ a -0.5) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((y + x) <= -4e-113) {
		tmp = fma(z, t_1, fma(b, (a + -0.5), x));
	} else {
		tmp = fma(z, t_1, fma(b, (a + -0.5), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(y + x) <= -4e-113)
		tmp = fma(z, t_1, fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, t_1, fma(b, Float64(a + -0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + x), $MachinePrecision], -4e-113], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + N[(b * N[(a + -0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -3.99999999999999991e-113

   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 y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

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

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

   5. Simplified 77.0%

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

   if -3.99999999999999991e-113 < (+.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. log-rec N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. log-rec N/A

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

      14. sub-neg N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 83.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -4 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(b, a + -0.5, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+159}:\\ \;\;\;\;y + z \cdot t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+161}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= z -3.2e+159)
     (+ y (* z t_1))
     (if (<= z 1.75e+161) (+ y (fma b (+ a -0.5) x)) (fma z t_1 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (z <= -3.2e+159) {
		tmp = y + (z * t_1);
	} else if (z <= 1.75e+161) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = fma(z, t_1, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (z <= -3.2e+159)
		tmp = Float64(y + Float64(z * t_1));
	elseif (z <= 1.75e+161)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = fma(z, t_1, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+159], N[(y + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+161], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z * t$95$1 + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999985e159

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf

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

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

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

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

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

      3. log-lowering-log.f64 77.7

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

   8. Simplified 77.7%

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

   if -3.19999999999999985e159 < z < 1.74999999999999994e161

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 90.9

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

   5. Simplified 90.9%

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

   if 1.74999999999999994e161 < 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. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 73.9

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

   5. Simplified 73.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. log-lowering-log.f64 64.4

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

   8. Simplified 64.4%

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

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, y\right)\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+159}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) y)))
   (if (<= z -4.3e+158)
     t_1
     (if (<= z 1.22e+159) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), y);
	double tmp;
	if (z <= -4.3e+158) {
		tmp = t_1;
	} else if (z <= 1.22e+159) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), y)
	tmp = 0.0
	if (z <= -4.3e+158)
		tmp = t_1;
	elseif (z <= 1.22e+159)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -4.3e+158], t$95$1, If[LessEqual[z, 1.22e+159], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3e158 or 1.22000000000000004e159 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. log-lowering-log.f64 71.1

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

   8. Simplified 71.1%

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

   if -4.3e158 < z < 1.22000000000000004e159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 90.9

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

   5. Simplified 90.9%

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

Alternative 9: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 1 - \log t, x\right)\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+123}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- 1.0 (log t)) x)))
   (if (<= z -4.3e+158)
     t_1
     (if (<= z 2.05e+123) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (1.0 - log(t)), x);
	double tmp;
	if (z <= -4.3e+158) {
		tmp = t_1;
	} else if (z <= 2.05e+123) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(1.0 - log(t)), x)
	tmp = 0.0
	if (z <= -4.3e+158)
		tmp = t_1;
	elseif (z <= 2.05e+123)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4.3e+158], t$95$1, If[LessEqual[z, 2.05e+123], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3e158 or 2.04999999999999995e123 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

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

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

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

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. log-lowering-log.f64 68.6

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

   8. Simplified 68.6%

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

   if -4.3e158 < z < 2.04999999999999995e123

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 91.3

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

   5. Simplified 91.3%

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

Alternative 10: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+161}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a + -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- 1.0 (log t)))))
   (if (<= z -6.6e+202)
     t_1
     (if (<= z 5.6e+161) (+ y (fma b (+ a -0.5) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - log(t));
	double tmp;
	if (z <= -6.6e+202) {
		tmp = t_1;
	} else if (z <= 5.6e+161) {
		tmp = y + fma(b, (a + -0.5), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(1.0 - log(t)))
	tmp = 0.0
	if (z <= -6.6e+202)
		tmp = t_1;
	elseif (z <= 5.6e+161)
		tmp = Float64(y + fma(b, Float64(a + -0.5), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+202], t$95$1, If[LessEqual[z, 5.6e+161], N[(y + N[(b * N[(a + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5999999999999998e202 or 5.60000000000000041e161 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

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

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

      7. log-lowering-log.f64 65.6

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

   5. Simplified 65.6%

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

   if -6.5999999999999998e202 < z < 5.60000000000000041e161

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 89.4

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

   5. Simplified 89.4%

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

Alternative 11: 59.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+179}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+265)
     (* b a)
     (if (<= t_1 -2e+186)
       (fma b -0.5 x)
       (if (<= t_1 2e+179) (+ y x) (fma b a x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = b * a;
	} else if (t_1 <= -2e+186) {
		tmp = fma(b, -0.5, x);
	} else if (t_1 <= 2e+179) {
		tmp = y + x;
	} else {
		tmp = fma(b, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+265)
		tmp = Float64(b * a);
	elseif (t_1 <= -2e+186)
		tmp = fma(b, -0.5, x);
	elseif (t_1 <= 2e+179)
		tmp = Float64(y + x);
	else
		tmp = fma(b, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+265], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -2e+186], N[(b * -0.5 + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+179], N[(y + x), $MachinePrecision], N[(b * a + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000013e265

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 83.9

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

   5. Simplified 83.9%

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

   if -2.00000000000000013e265 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999996e186

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 68.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 62.2

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

   8. Simplified 62.2%

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

   if -1.99999999999999996e186 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999996e179

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 53.9%

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

   if 1.99999999999999996e179 < (*.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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. --lowering--.f64 N/A

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

      17. log-lowering-log.f64 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 72.8

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

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+179}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+265)
     (* b a)
     (if (<= t_1 -2e+186)
       (fma b -0.5 x)
       (if (<= t_1 5e+259) (+ y x) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = b * a;
	} else if (t_1 <= -2e+186) {
		tmp = fma(b, -0.5, x);
	} else if (t_1 <= 5e+259) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+265)
		tmp = Float64(b * a);
	elseif (t_1 <= -2e+186)
		tmp = fma(b, -0.5, x);
	elseif (t_1 <= 5e+259)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+265], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -2e+186], N[(b * -0.5 + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+259], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000013e265 or 5.00000000000000033e259 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.1

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

   5. Simplified 91.1%

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

   if -2.00000000000000013e265 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999996e186

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 68.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 62.2

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

   8. Simplified 62.2%

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

   if -1.99999999999999996e186 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.00000000000000033e259

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 52.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, x\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+259}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+191}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))) (t_2 (* b (+ a -0.5))))
   (if (<= t_1 -2e+191) t_2 (if (<= t_1 5e+211) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -2e+191) {
		tmp = t_2;
	} else if (t_1 <= 5e+211) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    t_2 = b * (a + (-0.5d0))
    if (t_1 <= (-2d+191)) then
        tmp = t_2
    else if (t_1 <= 5d+211) then
        tmp = y + x
    else
        tmp = t_2
    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 = b * (a - 0.5);
	double t_2 = b * (a + -0.5);
	double tmp;
	if (t_1 <= -2e+191) {
		tmp = t_2;
	} else if (t_1 <= 5e+211) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	t_2 = b * (a + -0.5)
	tmp = 0
	if t_1 <= -2e+191:
		tmp = t_2
	elif t_1 <= 5e+211:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	t_2 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (t_1 <= -2e+191)
		tmp = t_2;
	elseif (t_1 <= 5e+211)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	t_2 = b * (a + -0.5);
	tmp = 0.0;
	if (t_1 <= -2e+191)
		tmp = t_2;
	elseif (t_1 <= 5e+211)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+191], t$95$2, If[LessEqual[t$95$1, 5e+211], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000015e191 or 4.9999999999999995e211 < (*.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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 86.3

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

   5. Simplified 86.3%

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

   if -2.00000000000000015e191 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999995e211

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 53.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+191}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+211}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -2e+265) (* b a) (if (<= t_1 5e+259) (+ y x) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = b * a;
	} else if (t_1 <= 5e+259) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (a - 0.5d0)
    if (t_1 <= (-2d+265)) then
        tmp = b * a
    else if (t_1 <= 5d+259) then
        tmp = y + x
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = b * a;
	} else if (t_1 <= 5e+259) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -2e+265:
		tmp = b * a
	elif t_1 <= 5e+259:
		tmp = y + x
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -2e+265)
		tmp = Float64(b * a);
	elseif (t_1 <= 5e+259)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -2e+265)
		tmp = b * a;
	elseif (t_1 <= 5e+259)
		tmp = y + x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+265], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 5e+259], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000013e265 or 5.00000000000000033e259 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.1

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

   5. Simplified 91.1%

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

   if -2.00000000000000013e265 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.00000000000000033e259

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 50.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 5 \cdot 10^{+259}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x\right)\\ \mathbf{elif}\;y + x \leq 10^{-82}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ y x) -4e-49)
   (fma b a x)
   (if (<= (+ y x) 1e-82) (* b (+ a -0.5)) (+ y (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y + x) <= -4e-49) {
		tmp = fma(b, a, x);
	} else if ((y + x) <= 1e-82) {
		tmp = b * (a + -0.5);
	} else {
		tmp = y + (b * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y + x) <= -4e-49)
		tmp = fma(b, a, x);
	elseif (Float64(y + x) <= 1e-82)
		tmp = Float64(b * Float64(a + -0.5));
	else
		tmp = Float64(y + Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y + x), $MachinePrecision], -4e-49], N[(b * a + x), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 1e-82], N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision], N[(y + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -3.99999999999999975e-49

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-in N/A

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

      10. log-rec N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. --lowering--.f64 N/A

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

      17. log-lowering-log.f64 70.8

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

   8. Simplified 70.8%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 47.4

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

   11. Simplified 47.4%

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

   if -3.99999999999999975e-49 < (+.f64 x y) < 1e-82

   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 inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 57.6

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

   5. Simplified 57.6%

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

   if 1e-82 < (+.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

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate--l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. log-rec N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate-+r+ N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 50.4

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

   8. Simplified 50.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x\right)\\ \mathbf{elif}\;y + x \leq 10^{-82}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;b \leq -4.7 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+179}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (+ a -0.5))))
   (if (<= b -4.7e+212) t_1 (if (<= b 6.6e+179) (+ x (fma b a y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a + -0.5);
	double tmp;
	if (b <= -4.7e+212) {
		tmp = t_1;
	} else if (b <= 6.6e+179) {
		tmp = x + fma(b, a, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a + -0.5))
	tmp = 0.0
	if (b <= -4.7e+212)
		tmp = t_1;
	elseif (b <= 6.6e+179)
		tmp = Float64(x + fma(b, a, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.7e+212], t$95$1, If[LessEqual[b, 6.6e+179], N[(x + N[(b * a + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.69999999999999991e212 or 6.59999999999999955e179 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 87.1

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

   5. Simplified 87.1%

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

   if -4.69999999999999991e212 < b < 6.59999999999999955e179

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 65.6

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

   8. Simplified 65.6%

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

Alternative 17: 65.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.15e-81) (fma (+ a -0.5) b x) (+ x (fma b a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.15e-81) {
		tmp = fma((a + -0.5), b, x);
	} else {
		tmp = x + fma(b, a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.15e-81)
		tmp = fma(Float64(a + -0.5), b, x);
	else
		tmp = Float64(x + fma(b, a, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.15e-81], N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(x + N[(b * a + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999996e-81

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 58.1%

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 58.1

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

   7. Applied egg-rr 58.1%

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

   if 1.14999999999999996e-81 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 66.9

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

   8. Simplified 66.9%

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

Alternative 18: 78.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-lowering-+.f64 74.1

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

5. Simplified 74.1%

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

Alternative 19: 42.2% 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 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. associate--l+ N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. log-rec N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. associate-+r+ N/A

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

5. Simplified 99.9%

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

6. Taylor expanded in x around inf

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

8. Simplified 41.9%

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

Alternative 20: 22.1% accurate, 126.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 19.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(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)))