Average Error: 16.0 → 0.5
Time: 18.6s
Precision: binary32
Cost: 3680
\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x + \left(\frac{\frac{\frac{-0.125}{x}}{x}}{x} + \left(x + \frac{-0.5}{x}\right)\right)\right) \]
(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary32
 (log (+ x (+ (/ (/ (/ -0.125 x) x) x) (+ x (/ -0.5 x))))))
float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}
↓
float code(float x) {
	return logf((x + ((((-0.125f / x) / x) / x) + (x + (-0.5f / x)))));
}

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function
↓
real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (((((-0.125e0) / x) / x) / x) + (x + ((-0.5e0) / x)))))
end function

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end
↓
function code(x)
	return log(Float32(x + Float32(Float32(Float32(Float32(Float32(-0.125) / x) / x) / x) + Float32(x + Float32(Float32(-0.5) / x)))))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end
↓
function tmp = code(x)
	tmp = log((x + ((((single(-0.125) / x) / x) / x) + (x + (single(-0.5) / x)))));
end

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\frac{\frac{\frac{-0.125}{x}}{x}}{x} + \left(x + \frac{-0.5}{x}\right)\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.0
Target0.3
Herbie0.5
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation

  1. Initial program 16.0

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

  2. Taylor expanded in x around inf 0.5

    \[\leadsto \log \left(x + \color{blue}{\left(\left(-0.125 \cdot {\left(\frac{1}{x}\right)}^{3} + x\right) - 0.5 \cdot \frac{1}{x}\right)}\right) \]

  3. Simplified0.5

    \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{-0.5}{x}\right) + \frac{-0.125}{{x}^{3}}\right)}\right) \]
    Proof
    (+.f32 (+.f32 x (/.f32 -1/2 x)) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (+.f32 x (/.f32 (Rewrite<= metadata-eval (neg.f32 1/2)) x)) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (+.f32 x (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1/2 x)))) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (+.f32 x (neg.f32 (/.f32 (Rewrite<= metadata-eval (*.f32 1/2 1)) x))) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (+.f32 x (neg.f32 (Rewrite<= associate-*r/_binary32 (*.f32 1/2 (/.f32 1 x))))) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 x (*.f32 1/2 (/.f32 1 x)))) (/.f32 -1/8 (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (-.f32 x (*.f32 1/2 (/.f32 1 x))) (/.f32 (Rewrite<= metadata-eval (*.f32 -1/8 1)) (pow.f32 x 3))): 0 points increase in error, 0 points decrease in error
    (+.f32 (-.f32 x (*.f32 1/2 (/.f32 1 x))) (Rewrite<= associate-*r/_binary32 (*.f32 -1/8 (/.f32 1 (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
    (+.f32 (-.f32 x (*.f32 1/2 (/.f32 1 x))) (*.f32 -1/8 (/.f32 (Rewrite<= metadata-eval (pow.f32 1 3)) (pow.f32 x 3)))): 0 points increase in error, 0 points decrease in error
    (+.f32 (-.f32 x (*.f32 1/2 (/.f32 1 x))) (*.f32 -1/8 (Rewrite<= cube-div_binary32 (pow.f32 (/.f32 1 x) 3)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 -1/8 (pow.f32 (/.f32 1 x) 3)) (-.f32 x (*.f32 1/2 (/.f32 1 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (*.f32 -1/8 (pow.f32 (/.f32 1 x) 3)) x) (*.f32 1/2 (/.f32 1 x)))): 1 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.5

    \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) + \color{blue}{\frac{0.25}{x \cdot x} \cdot \frac{-0.5}{x}}\right)\right) \]

  5. Applied egg-rr0.5

    \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) + \color{blue}{\frac{\frac{\frac{-0.125}{x}}{x}}{x}}\right)\right) \]

  6. Final simplification0.5

    \[\leadsto \log \left(x + \left(\frac{\frac{\frac{-0.125}{x}}{x}}{x} + \left(x + \frac{-0.5}{x}\right)\right)\right) \]

Alternatives

Alternative 1
Error0.6
Cost3424
\[\log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]
Alternative 2
Error0.6
Cost3424
\[\log \left(x \cdot 2 + \frac{-0.5}{x}\right) \]
Alternative 3
Error0.8
Cost3328
\[-\log \left(\frac{0.5}{x}\right) \]
Alternative 4
Error1.1
Cost3296
\[\log \left(x + x\right) \]
Alternative 5
Error30.0
Cost32
\[0 \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))