Average Error: 62.0 → 49.1
Time: 9.9s
Precision: binary64
Cost: 7040
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}^{2} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (pow (+ 1.0 (* -0.3333333333333333 (/ (- x hi) 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 pow((1.0 + (-0.3333333333333333 * ((x - hi) / 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 + ((-0.3333333333333333d0) * ((x - 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 Math.pow((1.0 + (-0.3333333333333333 * ((x - hi) / lo))), 2.0);
}

def code(lo, hi, x):
	return (x - lo) / (hi - lo)
↓
def code(lo, hi, x):
	return math.pow((1.0 + (-0.3333333333333333 * ((x - hi) / 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(-0.3333333333333333 * Float64(Float64(x - 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 + (-0.3333333333333333 * ((x - hi) / lo))) ^ 2.0;
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(-0.3333333333333333 * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}^{2}

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. Simplified62.0

    \[\leadsto \color{blue}{\frac{lo - x}{lo - hi}} \]
    Proof
    (/.f64 (-.f64 lo x) (-.f64 lo hi)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 lo x) (-.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 lo x) (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 lo x)) (*.f64 -1 (-.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 lo x))) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 lo x))) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 lo) x)) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 lo)) x) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 lo))) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x lo)) (*.f64 -1 (-.f64 lo hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 lo) hi))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 lo)) hi)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (Rewrite<= +-commutative_binary64 (+.f64 hi (neg.f64 lo)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 x lo) (Rewrite<= sub-neg_binary64 (-.f64 hi lo))): 0 points increase in error, 0 points decrease in error

  3. Taylor expanded in lo around inf 57.9

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

  4. Simplified57.9

    \[\leadsto \color{blue}{1 - \frac{x - hi}{lo}} \]
    Proof
    (-.f64 1 (/.f64 (-.f64 x hi) lo)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x hi) lo)))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 x hi) lo)))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (-.f64 x hi)) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (/.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 x) (*.f64 -1 hi))) lo)): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 -1 x) lo) (/.f64 (*.f64 -1 hi) lo)))): 3 points increase in error, 0 points decrease in error
    (+.f64 1 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x lo))) (/.f64 (*.f64 -1 hi) lo))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (-.f64 (*.f64 -1 (/.f64 x lo)) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 hi lo))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (*.f64 -1 (/.f64 x lo))) (*.f64 -1 (/.f64 hi lo)))): 0 points increase in error, 2 points decrease in error
    (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x lo)) 1)) (*.f64 -1 (/.f64 hi lo))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr57.9

    \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \frac{x - hi}{lo}}\right)}^{2} \cdot \sqrt[3]{1 - \frac{x - hi}{lo}}} \]

  6. Taylor expanded in lo around inf 51.6

    \[\leadsto {\left(\sqrt[3]{1 - \frac{x - hi}{lo}}\right)}^{2} \cdot \color{blue}{1} \]

  7. Taylor expanded in lo around -inf 49.1

    \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{2} \cdot 1 \]

  8. Final simplification49.1

    \[\leadsto {\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}^{2} \]

Alternatives

Alternative 1
Error51.7
Cost576
\[1 + \frac{x - hi}{lo} \cdot -0.6666666666666666 \]
Alternative 2
Error52.0
Cost320
\[\frac{x - lo}{hi} \]
Alternative 3
Error52.0
Cost256
\[-\frac{lo}{hi} \]
Alternative 4
Error52.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022296 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))