Average Error: 62.0 → 51.5
Time: 11.0s
Precision: binary64
Cost: 20672
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[1 + \left(\frac{hi \cdot \sqrt{{\left(1 + \frac{hi}{lo}\right)}^{2}}}{lo} + x \cdot \left(\frac{-1}{lo} - \frac{hi}{{lo}^{2}}\right)\right) \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (+
   (/ (* hi (sqrt (pow (+ 1.0 (/ hi lo)) 2.0))) lo)
   (* x (- (/ -1.0 lo) (/ hi (pow lo 2.0)))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return 1.0 + (((hi * sqrt(pow((1.0 + (hi / lo)), 2.0))) / lo) + (x * ((-1.0 / lo) - (hi / pow(lo, 2.0)))));
}

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 = 1.0d0 + (((hi * sqrt(((1.0d0 + (hi / lo)) ** 2.0d0))) / lo) + (x * (((-1.0d0) / lo) - (hi / (lo ** 2.0d0)))))
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 1.0 + (((hi * Math.sqrt(Math.pow((1.0 + (hi / lo)), 2.0))) / lo) + (x * ((-1.0 / lo) - (hi / Math.pow(lo, 2.0)))));
}

def code(lo, hi, x):
	return (x - lo) / (hi - lo)
↓
def code(lo, hi, x):
	return 1.0 + (((hi * math.sqrt(math.pow((1.0 + (hi / lo)), 2.0))) / lo) + (x * ((-1.0 / lo) - (hi / math.pow(lo, 2.0)))))

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(Float64(hi * sqrt((Float64(1.0 + Float64(hi / lo)) ^ 2.0))) / lo) + Float64(x * Float64(Float64(-1.0 / lo) - Float64(hi / (lo ^ 2.0))))))
end

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end
↓
function tmp = code(lo, hi, x)
	tmp = 1.0 + (((hi * sqrt(((1.0 + (hi / lo)) ^ 2.0))) / lo) + (x * ((-1.0 / lo) - (hi / (lo ^ 2.0)))));
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi * N[Sqrt[N[Power[N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] + N[(x * N[(N[(-1.0 / lo), $MachinePrecision] - N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
1 + \left(\frac{hi \cdot \sqrt{{\left(1 + \frac{hi}{lo}\right)}^{2}}}{lo} + x \cdot \left(\frac{-1}{lo} - \frac{hi}{{lo}^{2}}\right)\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 lo around inf 64.0

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

  3. Simplified51.9

    \[\leadsto \color{blue}{1 + \frac{x - hi}{lo} \cdot \left(-1 - \frac{hi}{lo}\right)} \]
    Proof
    (+.f64 1 (*.f64 (/.f64 (-.f64 x hi) lo) (-.f64 -1 (/.f64 hi lo)))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (*.f64 (/.f64 hi lo) (/.f64 (-.f64 x hi) lo))))): 13 points increase in error, 13 points decrease in error
    (+.f64 1 (-.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 hi (-.f64 x hi)) (*.f64 lo lo))))): 256 points increase in error, 0 points decrease in error
    (+.f64 1 (-.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x hi) hi)) (*.f64 lo lo)))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (-.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (/.f64 (*.f64 (-.f64 x hi) hi) (Rewrite<= unpow2_binary64 (pow.f64 lo 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (neg.f64 (/.f64 (*.f64 (-.f64 x hi) hi) (pow.f64 lo 2)))))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (+.f64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 x hi) hi) (pow.f64 lo 2)))))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 x hi) hi) (pow.f64 lo 2))) (*.f64 -1 (/.f64 (-.f64 x hi) lo))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 -1 (/.f64 (*.f64 (-.f64 x hi) hi) (pow.f64 lo 2)))) (*.f64 -1 (/.f64 (-.f64 x hi) lo)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 1 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (*.f64 (-.f64 x hi) hi)) (pow.f64 lo 2)))) (*.f64 -1 (/.f64 (-.f64 x hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 1 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -1 (-.f64 x hi)) hi)) (pow.f64 lo 2))) (*.f64 -1 (/.f64 (-.f64 x hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 1 (/.f64 (*.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) hi) (pow.f64 lo 2))) (*.f64 -1 (/.f64 (-.f64 x hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 1 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi)))) (pow.f64 lo 2))) (*.f64 -1 (/.f64 (-.f64 x hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) (pow.f64 lo 2)) 1)) (*.f64 -1 (/.f64 (-.f64 x hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 (/.f64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) (pow.f64 lo 2)) 1) (*.f64 -1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x lo) (/.f64 hi lo))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (+.f64 (/.f64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) (pow.f64 lo 2)) 1) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 x lo)) (*.f64 -1 (/.f64 hi lo))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (/.f64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) (pow.f64 lo 2)) 1) (*.f64 -1 (/.f64 x lo))) (*.f64 -1 (/.f64 hi lo)))): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x lo)) (+.f64 (/.f64 (*.f64 hi (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) (pow.f64 lo 2)) 1))) (*.f64 -1 (/.f64 hi lo))): 0 points increase in error, 0 points decrease in error

  4. Taylor expanded in x around 0 51.9

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

  5. Applied egg-rr51.5

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

  6. Final simplification51.5

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

Alternatives

Alternative 1
Error51.7
Cost13440
\[\sqrt{{\left(\frac{\frac{lo}{hi}}{\frac{hi}{x - lo}}\right)}^{2}} \]
Alternative 2
Error51.7
Cost13376
\[\sqrt{{\left(\frac{lo}{hi} \cdot \frac{-lo}{hi}\right)}^{2}} \]
Alternative 3
Error51.9
Cost1472
\[\left(1 - \frac{x}{lo}\right) + \left(\frac{1}{lo} - \frac{x}{lo \cdot lo}\right) \cdot \left(hi \cdot \left(1 + \frac{hi}{lo}\right)\right) \]
Alternative 4
Error51.9
Cost832
\[1 + \frac{-1 - \frac{hi}{lo}}{\frac{lo}{x - hi}} \]
Alternative 5
Error51.9
Cost704
\[1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
Alternative 6
Error52.0
Cost320
\[\frac{x - lo}{hi} \]
Alternative 7
Error52.0
Cost256
\[\frac{-lo}{hi} \]
Alternative 8
Error52.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022318 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))