xlohi (overflows)

Percentage Accurate: 3.1% → 64.5%

Time: 10.8s

Alternatives: 9

Speedup: 7.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := t_0 \cdot \left(1 - \frac{lo}{hi}\right)\\ t_2 := \frac{hi \cdot \frac{hi}{lo}}{x - lo}\\ t_3 := {t_2}^{-2}\\ t_4 := {t_0}^{2}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{{t_0}^{3} + {t_2}^{-3}}{\mathsf{fma}\left(t_0, t_1, t_3\right)}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{t_4 - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\mathsf{log1p}\left(t_4\right)} + -1\right) - t_3}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi))
        (t_1 (* t_0 (- 1.0 (/ lo hi))))
        (t_2 (/ (* hi (/ hi lo)) (- x lo)))
        (t_3 (pow t_2 -2.0))
        (t_4 (pow t_0 2.0)))
   (if (<= x -1.8e-16)
     (/ (+ (pow t_0 3.0) (pow t_2 -3.0)) (fma t_0 t_1 t_3))
     (if (<= x 1.45)
       (/ (- t_4 (pow (/ (* hi hi) (* x lo)) -2.0)) t_1)
       (/ (- (+ (exp (log1p t_4)) -1.0) t_3) t_1)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = t_0 * (1.0 - (lo / hi));
	double t_2 = (hi * (hi / lo)) / (x - lo);
	double t_3 = pow(t_2, -2.0);
	double t_4 = pow(t_0, 2.0);
	double tmp;
	if (x <= -1.8e-16) {
		tmp = (pow(t_0, 3.0) + pow(t_2, -3.0)) / fma(t_0, t_1, t_3);
	} else if (x <= 1.45) {
		tmp = (t_4 - pow(((hi * hi) / (x * lo)), -2.0)) / t_1;
	} else {
		tmp = ((exp(log1p(t_4)) + -1.0) - t_3) / t_1;
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = Float64(t_0 * Float64(1.0 - Float64(lo / hi)))
	t_2 = Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo))
	t_3 = t_2 ^ -2.0
	t_4 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -1.8e-16)
		tmp = Float64(Float64((t_0 ^ 3.0) + (t_2 ^ -3.0)) / fma(t_0, t_1, t_3));
	elseif (x <= 1.45)
		tmp = Float64(Float64(t_4 - (Float64(Float64(hi * hi) / Float64(x * lo)) ^ -2.0)) / t_1);
	else
		tmp = Float64(Float64(Float64(exp(log1p(t_4)) + -1.0) - t_3) / t_1);
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, -2.0], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -1.8e-16], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$2, -3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(t$95$4 - N[Power[N[(N[(hi * hi), $MachinePrecision] / N[(x * lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[Exp[N[Log[1 + t$95$4], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] - t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999991e-16

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.1%

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

      6. div-sub 9.1%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.1%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-times 9.1%

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

      4. metadata-eval 9.1%

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

   6. Applied egg-rr 9.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. flip3-+ 9.1%

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

      2. div-inv 9.1%

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

   8. Applied egg-rr 9.1%

      \[\leadsto \color{blue}{\left({\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{{\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2} + \frac{x - lo}{hi} \cdot \left(\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-*r/ 9.9%

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

      5. +-commutative 9.9%

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

      6. fma-def 9.9%

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

      7. *-rgt-identity 9.9%

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

      8. distribute-lft-out-- 9.9%

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

   10. Simplified 34.1%

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

   if -1.79999999999999991e-16 < x < 1.44999999999999996

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.1%

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

      6. div-sub 9.1%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.1%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-times 9.1%

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

      4. metadata-eval 9.1%

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

   6. Applied egg-rr 9.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 9.1%

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

      2. flip-+ 9.1%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 9.1%

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

      4. inv-pow 9.1%

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

      5. inv-pow 9.1%

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

      6. pow-prod-up 9.1%

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

      7. associate-*l/ 29.4%

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

      8. *-un-lft-identity 29.4%

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

      9. times-frac 9.1%

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

      10. /-rgt-identity 9.1%

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

      11. metadata-eval 9.1%

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

   8. Applied egg-rr 9.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 29.6%

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

      2. *-rgt-identity 29.6%

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

      3. distribute-lft-out-- 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in lo around 0 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{{hi}^{2}}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   12. Step-by-step derivation

       1. unpow2 99.2%

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

   13. Simplified 99.2%

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

   if 1.44999999999999996 < x

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.8%

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

      6. div-sub 8.8%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.8%

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

      2. clear-num 8.8%

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

      3. frac-times 8.8%

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

      4. metadata-eval 8.8%

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

   6. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.8%

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

      2. flip-+ 8.8%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.8%

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

      4. inv-pow 8.8%

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

      5. inv-pow 8.8%

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

      6. pow-prod-up 8.8%

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

      7. associate-*l/ 27.6%

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

      8. *-un-lft-identity 27.6%

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

      9. times-frac 8.8%

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

      10. /-rgt-identity 8.8%

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

      11. metadata-eval 8.8%

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

   8. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 27.7%

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

      2. *-rgt-identity 27.7%

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

      3. distribute-lft-out-- 27.6%

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

   10. Simplified 27.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 27.6%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

       2. expm1-udef 27.6%

          \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} - 1\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   12. Applied egg-rr 27.6%

       \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} - 1\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-3}}{\mathsf{fma}\left(\frac{x - lo}{hi}, \frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right), {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}\right)}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} + -1\right) - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 2: 65.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}\\ t_1 := \frac{x - lo}{hi}\\ t_2 := t_1 \cdot \left(1 - \frac{lo}{hi}\right)\\ t_3 := {t_1}^{2}\\ \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(1 + \mathsf{expm1}\left(t_3\right)\right) - t_0}{t_2}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{t_3 - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\mathsf{log1p}\left(t_3\right)} + -1\right) - t_0}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ (* hi (/ hi lo)) (- x lo)) -2.0))
        (t_1 (/ (- x lo) hi))
        (t_2 (* t_1 (- 1.0 (/ lo hi))))
        (t_3 (pow t_1 2.0)))
   (if (<= x -1.4)
     (/ (- (log (+ 1.0 (expm1 t_3))) t_0) t_2)
     (if (<= x 1.35)
       (/ (- t_3 (pow (/ (* hi hi) (* x lo)) -2.0)) t_2)
       (/ (- (+ (exp (log1p t_3)) -1.0) t_0) t_2)))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (log((1.0 + expm1(t_3))) - t_0) / t_2;
	} else if (x <= 1.35) {
		tmp = (t_3 - pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((exp(log1p(t_3)) + -1.0) - t_0) / t_2;
	}
	return tmp;
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = Math.pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = Math.pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (Math.log((1.0 + Math.expm1(t_3))) - t_0) / t_2;
	} else if (x <= 1.35) {
		tmp = (t_3 - Math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((Math.exp(Math.log1p(t_3)) + -1.0) - t_0) / t_2;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)
	t_1 = (x - lo) / hi
	t_2 = t_1 * (1.0 - (lo / hi))
	t_3 = math.pow(t_1, 2.0)
	tmp = 0
	if x <= -1.4:
		tmp = (math.log((1.0 + math.expm1(t_3))) - t_0) / t_2
	elif x <= 1.35:
		tmp = (t_3 - math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2
	else:
		tmp = ((math.exp(math.log1p(t_3)) + -1.0) - t_0) / t_2
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo)) ^ -2.0
	t_1 = Float64(Float64(x - lo) / hi)
	t_2 = Float64(t_1 * Float64(1.0 - Float64(lo / hi)))
	t_3 = t_1 ^ 2.0
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(log(Float64(1.0 + expm1(t_3))) - t_0) / t_2);
	elseif (x <= 1.35)
		tmp = Float64(Float64(t_3 - (Float64(Float64(hi * hi) / Float64(x * lo)) ^ -2.0)) / t_2);
	else
		tmp = Float64(Float64(Float64(exp(log1p(t_3)) + -1.0) - t_0) / t_2);
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[x, -1.4], N[(N[(N[Log[N[(1.0 + N[(Exp[t$95$3] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.35], N[(N[(t$95$3 - N[Power[N[(N[(hi * hi), $MachinePrecision] / N[(x * lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[Exp[N[Log[1 + t$95$3], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.9%

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

      6. div-sub 8.9%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.9%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.9%

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

      2. clear-num 8.9%

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

      3. frac-times 8.9%

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

      4. metadata-eval 8.9%

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

   6. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.9%

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

      2. flip-+ 8.9%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.9%

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

      4. inv-pow 8.9%

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

      5. inv-pow 8.9%

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

      6. pow-prod-up 8.9%

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

      7. associate-*l/ 32.9%

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

      8. *-un-lft-identity 32.9%

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

      9. times-frac 8.9%

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

      10. /-rgt-identity 8.9%

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

      11. metadata-eval 8.9%

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

   8. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 32.9%

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

      2. *-rgt-identity 32.9%

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

      3. distribute-lft-out-- 32.9%

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

   10. Simplified 32.9%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 32.9%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

       2. log1p-udef 32.9%

          \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   12. Applied egg-rr 32.9%

       \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   if -1.3999999999999999 < x < 1.3500000000000001

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.2%

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

      6. div-sub 9.2%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.2%

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

      2. clear-num 9.2%

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

      3. frac-times 9.2%

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

      4. metadata-eval 9.2%

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

   6. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 9.2%

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

      2. flip-+ 9.2%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 9.2%

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

      4. inv-pow 9.2%

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

      5. inv-pow 9.2%

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

      6. pow-prod-up 9.2%

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

      7. associate-*l/ 29.9%

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

      8. *-un-lft-identity 29.9%

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

      9. times-frac 9.2%

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

      10. /-rgt-identity 9.2%

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

      11. metadata-eval 9.2%

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

   8. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.1%

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

      2. *-rgt-identity 30.1%

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

      3. distribute-lft-out-- 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in lo around 0 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{{hi}^{2}}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   12. Step-by-step derivation

       1. unpow2 99.2%

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

   13. Simplified 99.2%

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

   if 1.3500000000000001 < x

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.8%

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

      6. div-sub 8.8%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.8%

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

      2. clear-num 8.8%

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

      3. frac-times 8.8%

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

      4. metadata-eval 8.8%

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

   6. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.8%

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

      2. flip-+ 8.8%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.8%

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

      4. inv-pow 8.8%

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

      5. inv-pow 8.8%

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

      6. pow-prod-up 8.8%

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

      7. associate-*l/ 27.6%

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

      8. *-un-lft-identity 27.6%

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

      9. times-frac 8.8%

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

      10. /-rgt-identity 8.8%

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

      11. metadata-eval 8.8%

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

   8. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 27.7%

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

      2. *-rgt-identity 27.7%

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

      3. distribute-lft-out-- 27.6%

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

   10. Simplified 27.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 27.6%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

       2. expm1-udef 27.6%

          \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} - 1\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   12. Applied egg-rr 27.6%

       \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} - 1\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right) - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\mathsf{log1p}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)} + -1\right) - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 3: 65.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}\\ t_1 := \frac{x - lo}{hi}\\ t_2 := t_1 \cdot \left(1 - \frac{lo}{hi}\right)\\ t_3 := {t_1}^{2}\\ \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(1 + \mathsf{expm1}\left(t_3\right)\right) - t_0}{t_2}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{t_3 - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - t_0}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ (* hi (/ hi lo)) (- x lo)) -2.0))
        (t_1 (/ (- x lo) hi))
        (t_2 (* t_1 (- 1.0 (/ lo hi))))
        (t_3 (pow t_1 2.0)))
   (if (<= x -1.4)
     (/ (- (log (+ 1.0 (expm1 t_3))) t_0) t_2)
     (if (<= x 1.35)
       (/ (- t_3 (pow (/ (* hi hi) (* x lo)) -2.0)) t_2)
       (/ (- (* t_1 t_1) t_0) t_2)))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (log((1.0 + expm1(t_3))) - t_0) / t_2;
	} else if (x <= 1.35) {
		tmp = (t_3 - pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((t_1 * t_1) - t_0) / t_2;
	}
	return tmp;
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = Math.pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = Math.pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (Math.log((1.0 + Math.expm1(t_3))) - t_0) / t_2;
	} else if (x <= 1.35) {
		tmp = (t_3 - Math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((t_1 * t_1) - t_0) / t_2;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)
	t_1 = (x - lo) / hi
	t_2 = t_1 * (1.0 - (lo / hi))
	t_3 = math.pow(t_1, 2.0)
	tmp = 0
	if x <= -1.4:
		tmp = (math.log((1.0 + math.expm1(t_3))) - t_0) / t_2
	elif x <= 1.35:
		tmp = (t_3 - math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2
	else:
		tmp = ((t_1 * t_1) - t_0) / t_2
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo)) ^ -2.0
	t_1 = Float64(Float64(x - lo) / hi)
	t_2 = Float64(t_1 * Float64(1.0 - Float64(lo / hi)))
	t_3 = t_1 ^ 2.0
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(log(Float64(1.0 + expm1(t_3))) - t_0) / t_2);
	elseif (x <= 1.35)
		tmp = Float64(Float64(t_3 - (Float64(Float64(hi * hi) / Float64(x * lo)) ^ -2.0)) / t_2);
	else
		tmp = Float64(Float64(Float64(t_1 * t_1) - t_0) / t_2);
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[x, -1.4], N[(N[(N[Log[N[(1.0 + N[(Exp[t$95$3] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.35], N[(N[(t$95$3 - N[Power[N[(N[(hi * hi), $MachinePrecision] / N[(x * lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.9%

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

      6. div-sub 8.9%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.9%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.9%

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

      2. clear-num 8.9%

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

      3. frac-times 8.9%

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

      4. metadata-eval 8.9%

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

   6. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.9%

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

      2. flip-+ 8.9%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.9%

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

      4. inv-pow 8.9%

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

      5. inv-pow 8.9%

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

      6. pow-prod-up 8.9%

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

      7. associate-*l/ 32.9%

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

      8. *-un-lft-identity 32.9%

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

      9. times-frac 8.9%

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

      10. /-rgt-identity 8.9%

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

      11. metadata-eval 8.9%

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

   8. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 32.9%

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

      2. *-rgt-identity 32.9%

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

      3. distribute-lft-out-- 32.9%

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

   10. Simplified 32.9%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 32.9%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

       2. log1p-udef 32.9%

          \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   12. Applied egg-rr 32.9%

       \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right)} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

   if -1.3999999999999999 < x < 1.3500000000000001

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.2%

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

      6. div-sub 9.2%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.2%

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

      2. clear-num 9.2%

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

      3. frac-times 9.2%

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

      4. metadata-eval 9.2%

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

   6. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 9.2%

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

      2. flip-+ 9.2%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 9.2%

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

      4. inv-pow 9.2%

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

      5. inv-pow 9.2%

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

      6. pow-prod-up 9.2%

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

      7. associate-*l/ 29.9%

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

      8. *-un-lft-identity 29.9%

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

      9. times-frac 9.2%

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

      10. /-rgt-identity 9.2%

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

      11. metadata-eval 9.2%

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

   8. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.1%

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

      2. *-rgt-identity 30.1%

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

      3. distribute-lft-out-- 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in lo around 0 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{{hi}^{2}}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   12. Step-by-step derivation

       1. unpow2 99.2%

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

   13. Simplified 99.2%

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

   if 1.3500000000000001 < x

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.8%

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

      6. div-sub 8.8%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.8%

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

      2. clear-num 8.8%

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

      3. frac-times 8.8%

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

      4. metadata-eval 8.8%

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

   6. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.8%

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

      2. flip-+ 8.8%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.8%

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

      4. inv-pow 8.8%

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

      5. inv-pow 8.8%

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

      6. pow-prod-up 8.8%

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

      7. associate-*l/ 27.6%

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

      8. *-un-lft-identity 27.6%

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

      9. times-frac 8.8%

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

      10. /-rgt-identity 8.8%

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

      11. metadata-eval 8.8%

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

   8. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 27.7%

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

      2. *-rgt-identity 27.7%

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

      3. distribute-lft-out-- 27.6%

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

   10. Simplified 27.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. unpow2 27.6%

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

   12. Applied egg-rr 27.6%

       \[\leadsto \frac{\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(1 + \mathsf{expm1}\left({\left(\frac{x - lo}{hi}\right)}^{2}\right)\right) - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 4: 65.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}\\ t_1 := \frac{x - lo}{hi}\\ t_2 := t_1 \cdot \left(1 - \frac{lo}{hi}\right)\\ t_3 := {t_1}^{2}\\ \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(e^{t_3}\right) - t_0}{t_2}\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{t_3 - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - t_0}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ (* hi (/ hi lo)) (- x lo)) -2.0))
        (t_1 (/ (- x lo) hi))
        (t_2 (* t_1 (- 1.0 (/ lo hi))))
        (t_3 (pow t_1 2.0)))
   (if (<= x -1.4)
     (/ (- (log (exp t_3)) t_0) t_2)
     (if (<= x 1.25)
       (/ (- t_3 (pow (/ (* hi hi) (* x lo)) -2.0)) t_2)
       (/ (- (* t_1 t_1) t_0) t_2)))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (log(exp(t_3)) - t_0) / t_2;
	} else if (x <= 1.25) {
		tmp = (t_3 - pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((t_1 * t_1) - t_0) / t_2;
	}
	return tmp;
}

Fortran

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((hi * (hi / lo)) / (x - lo)) ** (-2.0d0)
    t_1 = (x - lo) / hi
    t_2 = t_1 * (1.0d0 - (lo / hi))
    t_3 = t_1 ** 2.0d0
    if (x <= (-1.4d0)) then
        tmp = (log(exp(t_3)) - t_0) / t_2
    else if (x <= 1.25d0) then
        tmp = (t_3 - (((hi * hi) / (x * lo)) ** (-2.0d0))) / t_2
    else
        tmp = ((t_1 * t_1) - t_0) / t_2
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = Math.pow(((hi * (hi / lo)) / (x - lo)), -2.0);
	double t_1 = (x - lo) / hi;
	double t_2 = t_1 * (1.0 - (lo / hi));
	double t_3 = Math.pow(t_1, 2.0);
	double tmp;
	if (x <= -1.4) {
		tmp = (Math.log(Math.exp(t_3)) - t_0) / t_2;
	} else if (x <= 1.25) {
		tmp = (t_3 - Math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2;
	} else {
		tmp = ((t_1 * t_1) - t_0) / t_2;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)
	t_1 = (x - lo) / hi
	t_2 = t_1 * (1.0 - (lo / hi))
	t_3 = math.pow(t_1, 2.0)
	tmp = 0
	if x <= -1.4:
		tmp = (math.log(math.exp(t_3)) - t_0) / t_2
	elif x <= 1.25:
		tmp = (t_3 - math.pow(((hi * hi) / (x * lo)), -2.0)) / t_2
	else:
		tmp = ((t_1 * t_1) - t_0) / t_2
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo)) ^ -2.0
	t_1 = Float64(Float64(x - lo) / hi)
	t_2 = Float64(t_1 * Float64(1.0 - Float64(lo / hi)))
	t_3 = t_1 ^ 2.0
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(log(exp(t_3)) - t_0) / t_2);
	elseif (x <= 1.25)
		tmp = Float64(Float64(t_3 - (Float64(Float64(hi * hi) / Float64(x * lo)) ^ -2.0)) / t_2);
	else
		tmp = Float64(Float64(Float64(t_1 * t_1) - t_0) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = ((hi * (hi / lo)) / (x - lo)) ^ -2.0;
	t_1 = (x - lo) / hi;
	t_2 = t_1 * (1.0 - (lo / hi));
	t_3 = t_1 ^ 2.0;
	tmp = 0.0;
	if (x <= -1.4)
		tmp = (log(exp(t_3)) - t_0) / t_2;
	elseif (x <= 1.25)
		tmp = (t_3 - (((hi * hi) / (x * lo)) ^ -2.0)) / t_2;
	else
		tmp = ((t_1 * t_1) - t_0) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[x, -1.4], N[(N[(N[Log[N[Exp[t$95$3], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(t$95$3 - N[Power[N[(N[(hi * hi), $MachinePrecision] / N[(x * lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.9%

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

      6. div-sub 8.9%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.9%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.9%

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

      2. clear-num 8.9%

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

      3. frac-times 8.9%

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

      4. metadata-eval 8.9%

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

   6. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.9%

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

      2. flip-+ 8.9%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.9%

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

      4. inv-pow 8.9%

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

      5. inv-pow 8.9%

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

      6. pow-prod-up 8.9%

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

      7. associate-*l/ 32.9%

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

      8. *-un-lft-identity 32.9%

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

      9. times-frac 8.9%

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

      10. /-rgt-identity 8.9%

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

      11. metadata-eval 8.9%

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

   8. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 32.9%

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

      2. *-rgt-identity 32.9%

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

      3. distribute-lft-out-- 32.9%

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

   10. Simplified 32.9%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. add-log-exp 32.9%

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

   12. Applied egg-rr 32.9%

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

   if -1.3999999999999999 < x < 1.25

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.2%

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

      6. div-sub 9.2%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.2%

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

      2. clear-num 9.2%

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

      3. frac-times 9.2%

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

      4. metadata-eval 9.2%

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

   6. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 9.2%

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

      2. flip-+ 9.2%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 9.2%

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

      4. inv-pow 9.2%

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

      5. inv-pow 9.2%

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

      6. pow-prod-up 9.2%

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

      7. associate-*l/ 29.9%

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

      8. *-un-lft-identity 29.9%

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

      9. times-frac 9.2%

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

      10. /-rgt-identity 9.2%

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

      11. metadata-eval 9.2%

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

   8. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.1%

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

      2. *-rgt-identity 30.1%

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

      3. distribute-lft-out-- 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in lo around 0 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{{hi}^{2}}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   12. Step-by-step derivation

       1. unpow2 99.2%

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

   13. Simplified 99.2%

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

   if 1.25 < x

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.8%

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

      6. div-sub 8.8%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.8%

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

      2. clear-num 8.8%

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

      3. frac-times 8.8%

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

      4. metadata-eval 8.8%

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

   6. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.8%

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

      2. flip-+ 8.8%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.8%

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

      4. inv-pow 8.8%

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

      5. inv-pow 8.8%

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

      6. pow-prod-up 8.8%

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

      7. associate-*l/ 27.6%

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

      8. *-un-lft-identity 27.6%

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

      9. times-frac 8.8%

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

      10. /-rgt-identity 8.8%

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

      11. metadata-eval 8.8%

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

   8. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 27.7%

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

      2. *-rgt-identity 27.7%

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

      3. distribute-lft-out-- 27.6%

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

   10. Simplified 27.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. unpow2 27.6%

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

   12. Applied egg-rr 27.6%

       \[\leadsto \frac{\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{x - lo}{hi}\right)}^{2}}\right) - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 5: 65.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := t_0 \cdot \left(1 - \frac{lo}{hi}\right)\\ \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{t_0 \cdot t_0 - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)) (t_1 (* t_0 (- 1.0 (/ lo hi)))))
   (if (or (<= x -1.55) (not (<= x 1.45)))
     (/ (- (* t_0 t_0) (pow (/ (* hi (/ hi lo)) (- x lo)) -2.0)) t_1)
     (/ (- (pow t_0 2.0) (pow (/ (* hi hi) (* x lo)) -2.0)) t_1))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = t_0 * (1.0 - (lo / hi));
	double tmp;
	if ((x <= -1.55) || !(x <= 1.45)) {
		tmp = ((t_0 * t_0) - pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / t_1;
	} else {
		tmp = (pow(t_0, 2.0) - pow(((hi * hi) / (x * lo)), -2.0)) / t_1;
	}
	return tmp;
}

Fortran

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - lo) / hi
    t_1 = t_0 * (1.0d0 - (lo / hi))
    if ((x <= (-1.55d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = ((t_0 * t_0) - (((hi * (hi / lo)) / (x - lo)) ** (-2.0d0))) / t_1
    else
        tmp = ((t_0 ** 2.0d0) - (((hi * hi) / (x * lo)) ** (-2.0d0))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = t_0 * (1.0 - (lo / hi));
	double tmp;
	if ((x <= -1.55) || !(x <= 1.45)) {
		tmp = ((t_0 * t_0) - Math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / t_1;
	} else {
		tmp = (Math.pow(t_0, 2.0) - Math.pow(((hi * hi) / (x * lo)), -2.0)) / t_1;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = t_0 * (1.0 - (lo / hi))
	tmp = 0
	if (x <= -1.55) or not (x <= 1.45):
		tmp = ((t_0 * t_0) - math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / t_1
	else:
		tmp = (math.pow(t_0, 2.0) - math.pow(((hi * hi) / (x * lo)), -2.0)) / t_1
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = Float64(t_0 * Float64(1.0 - Float64(lo / hi)))
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.45))
		tmp = Float64(Float64(Float64(t_0 * t_0) - (Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo)) ^ -2.0)) / t_1);
	else
		tmp = Float64(Float64((t_0 ^ 2.0) - (Float64(Float64(hi * hi) / Float64(x * lo)) ^ -2.0)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	t_1 = t_0 * (1.0 - (lo / hi));
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 1.45)))
		tmp = ((t_0 * t_0) - (((hi * (hi / lo)) / (x - lo)) ^ -2.0)) / t_1;
	else
		tmp = ((t_0 ^ 2.0) - (((hi * hi) / (x * lo)) ^ -2.0)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.45]], $MachinePrecision]], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[N[(N[(hi * hi), $MachinePrecision] / N[(x * lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1.44999999999999996 < x

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 8.9%

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

      6. div-sub 8.9%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 8.9%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 8.9%

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

      2. clear-num 8.9%

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

      3. frac-times 8.9%

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

      4. metadata-eval 8.9%

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

   6. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 8.9%

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

      2. flip-+ 8.9%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 8.9%

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

      4. inv-pow 8.9%

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

      5. inv-pow 8.9%

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

      6. pow-prod-up 8.9%

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

      7. associate-*l/ 29.8%

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

      8. *-un-lft-identity 29.8%

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

      9. times-frac 8.9%

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

      10. /-rgt-identity 8.9%

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

      11. metadata-eval 8.9%

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

   8. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 29.9%

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

      2. *-rgt-identity 29.9%

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

      3. distribute-lft-out-- 29.8%

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

   10. Simplified 29.8%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. unpow2 29.8%

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

   12. Applied egg-rr 29.8%

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

   if -1.55000000000000004 < x < 1.44999999999999996

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 9.2%

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

      6. div-sub 9.2%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 9.2%

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

      2. clear-num 9.2%

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

      3. frac-times 9.2%

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

      4. metadata-eval 9.2%

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

   6. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 9.2%

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

      2. flip-+ 9.2%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 9.2%

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

      4. inv-pow 9.2%

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

      5. inv-pow 9.2%

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

      6. pow-prod-up 9.2%

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

      7. associate-*l/ 29.9%

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

      8. *-un-lft-identity 29.9%

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

      9. times-frac 9.2%

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

      10. /-rgt-identity 9.2%

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

      11. metadata-eval 9.2%

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

   8. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.1%

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

      2. *-rgt-identity 30.1%

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

      3. distribute-lft-out-- 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in lo around 0 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{{hi}^{2}}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   12. Step-by-step derivation

       1. unpow2 99.2%

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

   13. Simplified 99.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{hi \cdot hi}{lo \cdot x}\right)}}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot hi}{x \cdot lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 6: 37.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;hi \leq 1.45875 \cdot 10^{+308}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{t_0 \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= hi 1.45875e+308)
     1.0
     (/
      (- (* t_0 t_0) (pow (/ (* hi (/ hi lo)) (- x lo)) -2.0))
      (* t_0 (- 1.0 (/ lo hi)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (hi <= 1.45875e+308) {
		tmp = 1.0;
	} else {
		tmp = ((t_0 * t_0) - pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / (t_0 * (1.0 - (lo / hi)));
	}
	return tmp;
}

Fortran

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
    real(8) :: tmp
    t_0 = (x - lo) / hi
    if (hi <= 1.45875d+308) then
        tmp = 1.0d0
    else
        tmp = ((t_0 * t_0) - (((hi * (hi / lo)) / (x - lo)) ** (-2.0d0))) / (t_0 * (1.0d0 - (lo / hi)))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (hi <= 1.45875e+308) {
		tmp = 1.0;
	} else {
		tmp = ((t_0 * t_0) - Math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / (t_0 * (1.0 - (lo / hi)));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if hi <= 1.45875e+308:
		tmp = 1.0
	else:
		tmp = ((t_0 * t_0) - math.pow(((hi * (hi / lo)) / (x - lo)), -2.0)) / (t_0 * (1.0 - (lo / hi)))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (hi <= 1.45875e+308)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(t_0 * t_0) - (Float64(Float64(hi * Float64(hi / lo)) / Float64(x - lo)) ^ -2.0)) / Float64(t_0 * Float64(1.0 - Float64(lo / hi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (hi <= 1.45875e+308)
		tmp = 1.0;
	else
		tmp = ((t_0 * t_0) - (((hi * (hi / lo)) / (x - lo)) ^ -2.0)) / (t_0 * (1.0 - (lo / hi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[hi, 1.45875e+308], 1.0, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.45875000000000008e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around inf 18.9%

      \[\leadsto \color{blue}{1} \]

   if 1.45875000000000008e308 < hi

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi}} \]
   3. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 13.0%

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

      6. div-sub 13.0%

         \[\leadsto \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

   4. Simplified 13.0%

      \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   5. Step-by-step derivation

      1. clear-num 13.0%

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

      2. clear-num 13.0%

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

      3. frac-times 13.0%

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

      4. metadata-eval 13.0%

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

   6. Applied egg-rr 13.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}} + \frac{x - lo}{hi} \]
   7. Step-by-step derivation

      1. +-commutative 13.0%

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

      2. flip-+ 13.0%

         \[\leadsto \color{blue}{\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}} \cdot \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}{\frac{x - lo}{hi} - \frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{lo}}}} \]

      3. pow2 13.0%

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

      4. inv-pow 13.0%

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

      5. inv-pow 13.0%

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

      6. pow-prod-up 13.0%

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

      7. associate-*l/ 56.1%

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

      8. *-un-lft-identity 56.1%

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

      9. times-frac 13.0%

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

      10. /-rgt-identity 13.0%

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

      11. metadata-eval 13.0%

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

   8. Applied egg-rr 13.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(hi \cdot \frac{\frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} - \frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 56.4%

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

      2. *-rgt-identity 56.4%

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

      3. distribute-lft-out-- 56.3%

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

   10. Simplified 56.3%

       \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]
   11. Step-by-step derivation

       1. unpow2 56.3%

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

   12. Applied egg-rr 56.3%

       \[\leadsto \frac{\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.45875 \cdot 10^{+308}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - {\left(\frac{hi \cdot \frac{hi}{lo}}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}\\ \end{array} \]

Alternative 7: 18.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

3. Final simplification 18.8%

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

Alternative 8: 18.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-lo}{hi} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = -lo / hi
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{lo}{hi}} \]
6. Step-by-step derivation

   1. neg-mul-1 18.8%

      \[\leadsto \color{blue}{-\frac{lo}{hi}} \]

   2. distribute-neg-frac 18.8%

      \[\leadsto \color{blue}{\frac{-lo}{hi}} \]

7. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{-lo}{hi}} \]

8. Final simplification 18.8%

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

Alternative 9: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

double code(double lo, double hi, double x) {
	return 1.0;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0;
}

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0;
end

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))