Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 7

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))

C

double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((1.0d0 / x) + (sqrt((1.0d0 - (x * x))) / x)))
end function

Java

public static double code(double x) {
	return Math.log(((1.0 / x) + (Math.sqrt((1.0 - (x * x))) / x)));
}

Python

def code(x):
	return math.log(((1.0 / x) + (math.sqrt((1.0 - (x * x))) / x)))

Julia

function code(x)
	return log(Float64(Float64(1.0 / x) + Float64(sqrt(Float64(1.0 - Float64(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))

C

double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((1.0d0 / x) + (sqrt((1.0d0 - (x * x))) / x)))
end function

Java

public static double code(double x) {
	return Math.log(((1.0 / x) + (Math.sqrt((1.0 - (x * x))) / x)));
}

Python

def code(x):
	return math.log(((1.0 / x) + (math.sqrt((1.0 - (x * x))) / x)))

Julia

function code(x)
	return log(Float64(Float64(1.0 / x) + Float64(sqrt(Float64(1.0 - Float64(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-1}{x} \cdot \left(-1 - \sqrt{\mathsf{fma}\left(-x, x, 1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (* (/ -1.0 x) (- -1.0 (sqrt (fma (- x) x 1.0))))))

C

double code(double x) {
	return log(((-1.0 / x) * (-1.0 - sqrt(fma(-x, x, 1.0)))));
}

Julia

function code(x)
	return log(Float64(Float64(-1.0 / x) * Float64(-1.0 - sqrt(fma(Float64(-x), x, 1.0)))))
end

Wolfram

code[x_] := N[Log[N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 - N[Sqrt[N[((-x) * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-/.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

   21. lift--.f64 N/A

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

   22. lift-*.f64 N/A

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

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

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

   24. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \log \left(\frac{-1}{x} \cdot \left(-1 - \sqrt{\mathsf{fma}\left(-x, x, 1\right)}\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)} + 1}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (+ (sqrt (fma (- x) x 1.0)) 1.0) x)))

C

double code(double x) {
	return log(((sqrt(fma(-x, x, 1.0)) + 1.0) / x));
}

Julia

function code(x)
	return log(Float64(Float64(sqrt(fma(Float64(-x), x, 1.0)) + 1.0) / x))
end

Wolfram

code[x_] := N[Log[N[(N[(N[Sqrt[N[((-x) * x + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-rgt1-in N/A

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

   6. lift-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 100.0

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \log \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)} + 1}{x}\right)} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x \cdot x, 0.125\right), x \cdot x, 0.5\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (log (* (fma (fma 0.0625 (* x x) 0.125) (* x x) 0.5) x))))

C

double code(double x) {
	return -log((fma(fma(0.0625, (x * x), 0.125), (x * x), 0.5) * x));
}

Julia

function code(x)
	return Float64(-log(Float64(fma(fma(0.0625, Float64(x * x), 0.125), Float64(x * x), 0.5) * x)))
end

Wolfram

code[x_] := (-N[Log[N[(N[(N[(0.0625 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-/.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

   21. lift--.f64 N/A

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

   22. lift-*.f64 N/A

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

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

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

   24. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{neg}\left(2\right)\right)}\right) \]

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 99.4

      \[\leadsto \log \left(\frac{-1}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, \color{blue}{-2}\right)\right) \]

7. Applied rewrites 99.4%

   \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right) \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. clear-num N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{-\log \left(\frac{x}{-\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto -\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto -\log \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto -\log \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

    3. +-commutative N/A

       \[\leadsto -\log \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x\right) \]

    4. *-commutative N/A

       \[\leadsto -\log \left(\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto -\log \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)} \cdot x\right) \]

    6. +-commutative N/A

       \[\leadsto -\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot {x}^{2} + \frac{1}{8}}, {x}^{2}, \frac{1}{2}\right) \cdot x\right) \]

    7. lower-fma.f64 N/A

       \[\leadsto -\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{16}, {x}^{2}, \frac{1}{8}\right)}, {x}^{2}, \frac{1}{2}\right) \cdot x\right) \]

    8. unpow2 N/A

       \[\leadsto -\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \color{blue}{x \cdot x}, \frac{1}{8}\right), {x}^{2}, \frac{1}{2}\right) \cdot x\right) \]

    9. lower-*.f64 N/A

       \[\leadsto -\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \color{blue}{x \cdot x}, \frac{1}{8}\right), {x}^{2}, \frac{1}{2}\right) \cdot x\right) \]

    10. unpow2 N/A

        \[\leadsto -\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{1}{8}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x\right) \]

    11. lower-*.f64 99.4

        \[\leadsto -\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x \cdot x, 0.125\right), \color{blue}{x \cdot x}, 0.5\right) \cdot x\right) \]

12. Applied rewrites 99.4%

    \[\leadsto -\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x \cdot x, 0.125\right), x \cdot x, 0.5\right) \cdot x\right)} \]
13. Add Preprocessing

Alternative 4: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\mathsf{fma}\left(x \cdot x, -0.5, 2\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (fma (* x x) -0.5 2.0) x)))

C

double code(double x) {
	return log((fma((x * x), -0.5, 2.0) / x));
}

Julia

function code(x)
	return log(Float64(fma(Float64(x * x), -0.5, 2.0) / x))
end

Wolfram

code[x_] := N[Log[N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.4

      \[\leadsto \log \left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 2\right)}{x}\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \log \color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, -0.5, 2\right)}{x}\right)} \]
6. Add Preprocessing

Alternative 5: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -\log \left(\mathsf{fma}\left(0.125, x \cdot x, 0.5\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log (* (fma 0.125 (* x x) 0.5) x))))

C

double code(double x) {
	return -log((fma(0.125, (x * x), 0.5) * x));
}

Julia

function code(x)
	return Float64(-log(Float64(fma(0.125, Float64(x * x), 0.5) * x)))
end

Wolfram

code[x_] := (-N[Log[N[(N[(0.125 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-/.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

   21. lift--.f64 N/A

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

   22. lift-*.f64 N/A

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

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

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

   24. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{neg}\left(2\right)\right)}\right) \]

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 99.4

      \[\leadsto \log \left(\frac{-1}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, \color{blue}{-2}\right)\right) \]

7. Applied rewrites 99.4%

   \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right) \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. clear-num N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{-\log \left(\frac{x}{-\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto -\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {x}^{2}\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto -\log \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto -\log \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right)} \]

    3. +-commutative N/A

       \[\leadsto -\log \left(\color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{2}\right)} \cdot x\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto -\log \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, \frac{1}{2}\right)} \cdot x\right) \]

    5. unpow2 N/A

       \[\leadsto -\log \left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x\right) \]

    6. lower-*.f64 99.4

       \[\leadsto -\log \left(\mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, 0.5\right) \cdot x\right) \]

12. Applied rewrites 99.4%

    \[\leadsto -\log \color{blue}{\left(\mathsf{fma}\left(0.125, x \cdot x, 0.5\right) \cdot x\right)} \]
13. Add Preprocessing

Alternative 6: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -\log \left(0.5 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log (* 0.5 x))))

C

double code(double x) {
	return -log((0.5 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log((0.5d0 * x))
end function

Java

public static double code(double x) {
	return -Math.log((0.5 * x));
}

Python

def code(x):
	return -math.log((0.5 * x))

Julia

function code(x)
	return Float64(-log(Float64(0.5 * x)))
end

MATLAB

function tmp = code(x)
	tmp = -log((0.5 * x));
end

Wolfram

code[x_] := (-N[Log[N[(0.5 * x), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-/.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

   21. lift--.f64 N/A

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

   22. lift-*.f64 N/A

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

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

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

   24. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{neg}\left(2\right)\right)}\right) \]

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 99.4

      \[\leadsto \log \left(\frac{-1}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, \color{blue}{-2}\right)\right) \]

7. Applied rewrites 99.4%

   \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right) \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. clear-num N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{-\log \left(\frac{x}{-\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto -\log \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
11. Step-by-step derivation

    1. lower-*.f64 99.2

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

12. Applied rewrites 99.2%

    \[\leadsto -\log \color{blue}{\left(0.5 \cdot x\right)} \]
13. Add Preprocessing

Alternative 7: 0.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(-0.5 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (* -0.5 x)))

C

double code(double x) {
	return log((-0.5 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((-0.5d0) * x))
end function

Java

public static double code(double x) {
	return Math.log((-0.5 * x));
}

Python

def code(x):
	return math.log((-0.5 * x))

Julia

function code(x)
	return log(Float64(-0.5 * x))
end

MATLAB

function tmp = code(x)
	tmp = log((-0.5 * x));
end

Wolfram

code[x_] := N[Log[N[(-0.5 * x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.4

      \[\leadsto \log \left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 2\right)}{x}\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \log \color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, -0.5, 2\right)}{x}\right)} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 0.0%

   \[\leadsto \log \left(-0.5 \cdot \color{blue}{x}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))