Average Error: 16.0 → 0.4
Time: 5.9s
Precision: binary32
Cost: 6688
\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[-\log \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right) \]
(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary32 (- (log (+ (/ 0.125 (pow x 3.0)) (/ 0.5 x)))))
float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}
↓
float code(float x) {
	return -logf(((0.125f / powf(x, 3.0f)) + (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(((0.125e0 / (x ** 3.0e0)) + (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 Float32(-log(Float32(Float32(Float32(0.125) / (x ^ Float32(3.0))) + 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(((single(0.125) / (x ^ single(3.0))) + (single(0.5) / x)));
end

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
-\log \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.0
Target0.3
Herbie0.4
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation

  1. Initial program 16.0

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

  2. Applied egg-rr29.8

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

  3. Simplified29.8

    \[\leadsto \color{blue}{-\log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]
    Proof
    (neg.f32 (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> neg-sub0_binary32 (-.f32 0 (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1)))))): 0 points increase in error, 0 points decrease in error
    (-.f32 (Rewrite<= metadata-eval (log.f32 1)) (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error
    (-.f32 (log.f32 (Rewrite<= metadata-eval (-.f32 0 -1))) (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error
    (-.f32 (log.f32 (-.f32 (Rewrite<= +-inverses_binary32 (-.f32 (*.f32 x x) (*.f32 x x))) -1)) (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error
    (-.f32 (log.f32 (Rewrite<= associate--r+_binary32 (-.f32 (*.f32 x x) (+.f32 (*.f32 x x) -1)))) (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error
    (-.f32 (log.f32 (-.f32 (*.f32 x x) (Rewrite<= fma-udef_binary32 (fma.f32 x x -1)))) (log.f32 (-.f32 x (sqrt.f32 (fma.f32 x x -1))))): 0 points increase in error, 0 points decrease in error

  4. Taylor expanded in x around inf 0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Alternatives

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

Error

Reproduce

herbie shell --seed 2022328 
(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)))))