Average Error: 62.0 → 51.7
Time: 7.6s
Precision: binary64
Cost: 13696
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\sqrt{{\left(\frac{lo}{hi}\right)}^{3} \cdot \left(\frac{lo}{hi} + \frac{x \cdot -2}{hi}\right)} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (sqrt (* (pow (/ lo hi) 3.0) (+ (/ lo hi) (/ (* x -2.0) hi)))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return sqrt((pow((lo / hi), 3.0) * ((lo / hi) + ((x * -2.0) / hi))));
}

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function
↓
real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = sqrt((((lo / hi) ** 3.0d0) * ((lo / hi) + ((x * (-2.0d0)) / hi))))
end function

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
public static double code(double lo, double hi, double x) {
	return Math.sqrt((Math.pow((lo / hi), 3.0) * ((lo / hi) + ((x * -2.0) / hi))));
}

def code(lo, hi, x):
	return (x - lo) / (hi - lo)
↓
def code(lo, hi, x):
	return math.sqrt((math.pow((lo / hi), 3.0) * ((lo / hi) + ((x * -2.0) / hi))))

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return sqrt(Float64((Float64(lo / hi) ^ 3.0) * Float64(Float64(lo / hi) + Float64(Float64(x * -2.0) / hi))))
end

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end
↓
function tmp = code(lo, hi, x)
	tmp = sqrt((((lo / hi) ^ 3.0) * ((lo / hi) + ((x * -2.0) / hi))));
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[Sqrt[N[(N[Power[N[(lo / hi), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] + N[(N[(x * -2.0), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
\sqrt{{\left(\frac{lo}{hi}\right)}^{3} \cdot \left(\frac{lo}{hi} + \frac{x \cdot -2}{hi}\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 62.0

    \[\frac{x - lo}{hi - lo} \]

  2. Taylor expanded in hi around inf 64.0

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

  3. Simplified58.1

    \[\leadsto \color{blue}{\left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi}} \]
    Proof
    (*.f64 (+.f64 (/.f64 lo hi) 1) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (/.f64 lo hi) (/.f64 (-.f64 x lo) hi)) (/.f64 (-.f64 x lo) hi))): 112 points increase in error, 99 points decrease in error
    (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 lo (-.f64 x lo)) (*.f64 hi hi))) (/.f64 (-.f64 x lo) hi)): 256 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (Rewrite<= unpow2_binary64 (pow.f64 hi 2))) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x hi) (/.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)) (/.f64 x hi)) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x hi) (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)))) (/.f64 lo hi)): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr52.5

    \[\leadsto \color{blue}{\sqrt{{\left(\frac{\frac{lo}{hi} + 1}{\frac{hi}{x - lo}}\right)}^{2}}} \]

  5. Taylor expanded in lo around inf 51.7

    \[\leadsto \sqrt{{\left(\frac{\color{blue}{\frac{lo}{hi}}}{\frac{hi}{x - lo}}\right)}^{2}} \]

  6. Taylor expanded in lo around inf 64.0

    \[\leadsto \sqrt{\color{blue}{\frac{{lo}^{4}}{{hi}^{4}} + -2 \cdot \frac{{lo}^{3} \cdot x}{{hi}^{4}}}} \]

  7. Simplified51.7

    \[\leadsto \sqrt{\color{blue}{{\left(\frac{lo}{hi}\right)}^{3} \cdot \left(\frac{x \cdot -2}{hi} + \frac{lo}{hi}\right)}} \]
    Proof
    (*.f64 (pow.f64 (/.f64 lo hi) 3) (+.f64 (/.f64 (*.f64 x -2) hi) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite=> cube-div_binary64 (/.f64 (pow.f64 lo 3) (pow.f64 hi 3))) (+.f64 (/.f64 (*.f64 x -2) hi) (/.f64 lo hi))): 256 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (pow.f64 lo 3) (Rewrite<= cube-unmult_binary64 (*.f64 hi (*.f64 hi hi)))) (+.f64 (/.f64 (*.f64 x -2) hi) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (pow.f64 lo 3) (*.f64 hi (Rewrite<= unpow2_binary64 (pow.f64 hi 2)))) (+.f64 (/.f64 (*.f64 x -2) hi) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))) (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 x)) hi) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 (*.f64 -2 x) hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2)))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2)))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 -2 x) (pow.f64 lo 3)) (*.f64 hi (*.f64 hi (pow.f64 hi 2))))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 (*.f64 -2 x) (pow.f64 lo 3)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 hi hi) (pow.f64 hi 2)))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 (*.f64 -2 x) (pow.f64 lo 3)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 hi 2)) (pow.f64 hi 2))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 (*.f64 -2 x) (pow.f64 lo 3)) (Rewrite=> pow-sqr_binary64 (pow.f64 hi (*.f64 2 2)))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 (*.f64 -2 x) (pow.f64 lo 3)) (pow.f64 hi (Rewrite=> metadata-eval 4))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 x (pow.f64 lo 3)))) (pow.f64 hi 4)) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (*.f64 -2 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 lo 3) x))) (pow.f64 hi 4)) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4)))) (*.f64 (/.f64 lo hi) (/.f64 (pow.f64 lo 3) (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 lo (pow.f64 lo 3)) (*.f64 hi (*.f64 hi (pow.f64 hi 2)))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 lo 3) lo)) (*.f64 hi (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (Rewrite=> pow-plus_binary64 (pow.f64 lo (+.f64 3 1))) (*.f64 hi (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (pow.f64 lo (Rewrite=> metadata-eval 4)) (*.f64 hi (*.f64 hi (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (pow.f64 lo 4) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 hi hi) (pow.f64 hi 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (pow.f64 lo 4) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 hi 2)) (pow.f64 hi 2)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (pow.f64 lo 4) (Rewrite=> pow-sqr_binary64 (pow.f64 hi (*.f64 2 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))) (/.f64 (pow.f64 lo 4) (pow.f64 hi (Rewrite=> metadata-eval 4)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 lo 4) (pow.f64 hi 4)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 lo 3) x) (pow.f64 hi 4))))): 0 points increase in error, 0 points decrease in error

  8. Final simplification51.7

    \[\leadsto \sqrt{{\left(\frac{lo}{hi}\right)}^{3} \cdot \left(\frac{lo}{hi} + \frac{x \cdot -2}{hi}\right)} \]

Alternatives

Alternative 1
Error51.7
Cost7616
\[\sqrt{\frac{\frac{lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} \]
Alternative 2
Error51.7
Cost448
\[\frac{lo}{hi} \cdot \frac{lo}{hi} \]
Alternative 3
Error52.0
Cost320
\[\frac{x - lo}{hi} \]
Alternative 4
Error52.0
Cost256
\[\frac{-lo}{hi} \]
Alternative 5
Error52.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))