xlohi (overflows)

Average Error: 62.0 → 0.5

Time: 10.8s

Precision: binary64

Cost: 15104

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{lo}{-hi} + 1\right) \cdot \left(t_0 \cdot t_0\right)} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/
    (pow t_0 3.0)
    (+
     (pow (* lo (/ (- x lo) (* hi hi))) 2.0)
     (* (+ (/ lo (- hi)) 1.0) (* t_0 t_0))))))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

↓

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return pow(t_0, 3.0) / (pow((lo * ((x - lo) / (hi * hi))), 2.0) + (((lo / -hi) + 1.0) * (t_0 * t_0)));
}

Fortran

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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = (t_0 ** 3.0d0) / (((lo * ((x - lo) / (hi * hi))) ** 2.0d0) + (((lo / -hi) + 1.0d0) * (t_0 * t_0)))
end function

Java

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) {
	double t_0 = (x - lo) / hi;
	return Math.pow(t_0, 3.0) / (Math.pow((lo * ((x - lo) / (hi * hi))), 2.0) + (((lo / -hi) + 1.0) * (t_0 * t_0)));
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

↓

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return math.pow(t_0, 3.0) / (math.pow((lo * ((x - lo) / (hi * hi))), 2.0) + (((lo / -hi) + 1.0) * (t_0 * t_0)))

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

↓

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64((t_0 ^ 3.0) / Float64((Float64(lo * Float64(Float64(x - lo) / Float64(hi * hi))) ^ 2.0) + Float64(Float64(Float64(lo / Float64(-hi)) + 1.0) * Float64(t_0 * t_0))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

↓

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = (t_0 ^ 3.0) / (((lo * ((x - lo) / (hi * hi))) ^ 2.0) + (((lo / -hi) + 1.0) * (t_0 * t_0)));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

↓

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(N[Power[N[(lo * N[(N[(x - lo), $MachinePrecision] / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(lo / (-hi)), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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. Simplified 57.9

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   Proof
   (+.f64 (*.f64 (/.f64 (-.f64 x lo) hi) (/.f64 lo hi)) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 x lo) lo) (*.f64 hi hi))) (/.f64 (-.f64 x lo) hi)): 1 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 lo (-.f64 x lo))) (*.f64 hi hi)) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 1 points decrease in error
   (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (Rewrite<= unpow2_binary64 (pow.f64 hi 2))) (/.f64 (-.f64 x lo) hi)): 1 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, 1 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))): 1 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, 1 points decrease in error

4. Applied egg-rr 0.5

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

5. Simplified 0.5

   \[\leadsto \color{blue}{\frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{lo}{-hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}}} \]
   Proof
   (/.f64 (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 x lo) (*.f64 hi hi)) lo)) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 x lo) lo) (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 3 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi)))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 19 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1)) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 19 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 x lo) (*.f64 hi hi)) lo)) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 23 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 x lo) lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 22 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi)))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 1 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 lo))) (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 17 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (neg.f64 lo))) (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 16 points increase in error, 3 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 (*.f64 -1 (neg.f64 lo)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 hi))) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 3 points increase in error, 16 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (neg.f64 lo) hi))) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 20 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (neg.f64 lo) hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 23 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (*.f64 1 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 lo) 1)) hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 23 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (*.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 lo) (/.f64 1 hi)))) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 22 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 1 (neg.f64 lo)) (/.f64 1 hi))) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 22 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (*.f64 (Rewrite=> *-lft-identity_binary64 (neg.f64 lo)) (/.f64 1 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 23 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (neg.f64 lo) 1) hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 23 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (neg.f64 lo)) hi) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 19 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 lo)) hi) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 3 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (*.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 lo hi))) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 7 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (pow.f64 (/.f64 (-.f64 x lo) hi) 2) (*.f64 (*.f64 -1 (/.f64 lo hi)) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))))): 19 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (+.f64 (pow.f64 (/.f64 (-.f64 x lo) hi) 2) (*.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 lo hi))) (pow.f64 (/.f64 (-.f64 x lo) hi) 2))))): 0 points increase in error, 16 points decrease in error
   (/.f64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) 1) (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 (/.f64 (-.f64 x lo) hi) 2) (*.f64 (/.f64 lo hi) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))))): 20 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 3) (pow.f64 (/.f64 (-.f64 x lo) hi) 3)) (/.f64 1 (+.f64 (pow.f64 (*.f64 (-.f64 x lo) (/.f64 lo (*.f64 hi hi))) 2) (-.f64 (pow.f64 (/.f64 (-.f64 x lo) hi) 2) (*.f64 (/.f64 lo hi) (pow.f64 (/.f64 (-.f64 x lo) hi) 2))))))): 0 points increase in error, 20 points decrease in error

6. Taylor expanded in hi around inf 64.0

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

7. Simplified 0.5

   \[\leadsto \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{lo}{-hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]
   Proof
   (/.f64 (pow.f64 (/.f64 (-.f64 x lo) hi) 3) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> cube-div_binary64 (/.f64 (pow.f64 (-.f64 x lo) 3) (pow.f64 hi 3))) (+.f64 (pow.f64 (*.f64 lo (/.f64 (-.f64 x lo) (*.f64 hi hi))) 2) (*.f64 (+.f64 (/.f64 lo (neg.f64 hi)) 1) (pow.f64 (/.f64 (-.f64 x lo) hi) 2)))): 0 points increase in error, 0 points decrease in error

8. Applied egg-rr 0.5

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

9. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	14656

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {t_0}^{2}}{t_0 \cdot \left(\frac{lo}{hi} + -1\right)} \end{array} \]

Alternative 2
Error	0.6
Cost	7744

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{1}{\frac{t_0 \cdot \left(\frac{lo}{hi} + -1\right)}{-{t_0}^{2}}} \end{array} \]

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	256

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

Alternative 5
Error	52.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))