xlohi (overflows)

Percentage Accurate: 3.1% → 22.4%

Time: 8.4s

Alternatives: 8

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: 22.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{hi}}\\ \mathbf{if}\;lo \leq -1.0458 \cdot 10^{+308}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{x}{hi \cdot hi} + \frac{-1}{hi}, lo \cdot t_0, t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2}}{\frac{\left(x - lo\right) + lo \cdot \frac{\mathsf{fma}\left(-1, x, lo\right)}{hi}}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (exp (/ x hi))))
   (if (<= lo -1.0458e+308)
     (log (fma (+ (/ x (* hi hi)) (/ -1.0 hi)) (* lo t_0) t_0))
     (/
      (- (pow (/ (- x lo) hi) 2.0) (pow (/ (- x lo) (* hi (/ hi lo))) 2.0))
      (/ (+ (- x lo) (* lo (/ (fma -1.0 x lo) hi))) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = exp((x / hi));
	double tmp;
	if (lo <= -1.0458e+308) {
		tmp = log(fma(((x / (hi * hi)) + (-1.0 / hi)), (lo * t_0), t_0));
	} else {
		tmp = (pow(((x - lo) / hi), 2.0) - pow(((x - lo) / (hi * (hi / lo))), 2.0)) / (((x - lo) + (lo * (fma(-1.0, x, lo) / hi))) / hi);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = exp(Float64(x / hi))
	tmp = 0.0
	if (lo <= -1.0458e+308)
		tmp = log(fma(Float64(Float64(x / Float64(hi * hi)) + Float64(-1.0 / hi)), Float64(lo * t_0), t_0));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / hi) ^ 2.0) - (Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0)) / Float64(Float64(Float64(x - lo) + Float64(lo * Float64(fma(-1.0, x, lo) / hi))) / hi));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lo, -1.0458e+308], N[Log[N[(N[(N[(x / N[(hi * hi), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / hi), $MachinePrecision]), $MachinePrecision] * N[(lo * t$95$0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x - lo), $MachinePrecision] + N[(lo * N[(N[(-1.0 * x + lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.04579999999999993e308

   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. Step-by-step derivation

      1. add-log-exp 18.8%

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

      2. associate-/r* 18.8%

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

      3. sub-div 18.8%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in lo around 0 20.6%

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

      1. fma-def 20.6%

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

      2. unpow2 20.6%

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

   9. Simplified 20.6%

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

   if -1.04579999999999993e308 < lo

   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 18.8%

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

      6. div-sub 18.8%

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

   4. Simplified 18.8%

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

      1. add-cube-cbrt 18.8%

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

      2. pow3 18.8%

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

   6. Applied egg-rr 18.8%

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

      1. rem-cbrt-cube 18.8%

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

      2. rem-cube-cbrt 18.8%

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

      3. flip-+ 18.8%

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

      4. frac-2neg 18.8%

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

   8. Applied egg-rr 15.0%

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

   10. Simplified 65.3%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.0458 \cdot 10^{+308}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{x}{hi \cdot hi} + \frac{-1}{hi}, lo \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2}}{\frac{\left(x - lo\right) + lo \cdot \frac{\mathsf{fma}\left(-1, x, lo\right)}{hi}}{hi}}\\ \end{array} \]

Alternative 2: 22.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;lo \leq -1.0458 \cdot 10^{+308}:\\ \;\;\;\;\log \left(e^{\frac{x}{hi}} \cdot \left(1 + lo \cdot \frac{\frac{x}{hi} + -1}{hi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2}}{\frac{\left(x - lo\right) + lo \cdot \frac{\mathsf{fma}\left(-1, x, lo\right)}{hi}}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= lo -1.0458e+308)
   (log (* (exp (/ x hi)) (+ 1.0 (* lo (/ (+ (/ x hi) -1.0) hi)))))
   (/
    (- (pow (/ (- x lo) hi) 2.0) (pow (/ (- x lo) (* hi (/ hi lo))) 2.0))
    (/ (+ (- x lo) (* lo (/ (fma -1.0 x lo) hi))) hi))))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (lo <= -1.0458e+308) {
		tmp = log((exp((x / hi)) * (1.0 + (lo * (((x / hi) + -1.0) / hi)))));
	} else {
		tmp = (pow(((x - lo) / hi), 2.0) - pow(((x - lo) / (hi * (hi / lo))), 2.0)) / (((x - lo) + (lo * (fma(-1.0, x, lo) / hi))) / hi);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (lo <= -1.0458e+308)
		tmp = log(Float64(exp(Float64(x / hi)) * Float64(1.0 + Float64(lo * Float64(Float64(Float64(x / hi) + -1.0) / hi)))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / hi) ^ 2.0) - (Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0)) / Float64(Float64(Float64(x - lo) + Float64(lo * Float64(fma(-1.0, x, lo) / hi))) / hi));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[lo, -1.0458e+308], N[Log[N[(N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(lo * N[(N[(N[(x / hi), $MachinePrecision] + -1.0), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x - lo), $MachinePrecision] + N[(lo * N[(N[(-1.0 * x + lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.04579999999999993e308

   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. Step-by-step derivation

      1. add-log-exp 18.8%

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

      2. associate-/r* 18.8%

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

      3. sub-div 18.8%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in lo around 0 20.6%

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

      1. associate-*r* 20.6%

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

      2. distribute-lft1-in 20.6%

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

   9. Simplified 20.6%

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

   if -1.04579999999999993e308 < lo

   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 18.8%

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

      6. div-sub 18.8%

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

   4. Simplified 18.8%

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

      1. add-cube-cbrt 18.8%

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

      2. pow3 18.8%

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

   6. Applied egg-rr 18.8%

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

      1. rem-cbrt-cube 18.8%

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

      2. rem-cube-cbrt 18.8%

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

      3. flip-+ 18.8%

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

      4. frac-2neg 18.8%

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

   8. Applied egg-rr 15.0%

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

   10. Simplified 65.3%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.0458 \cdot 10^{+308}:\\ \;\;\;\;\log \left(e^{\frac{x}{hi}} \cdot \left(1 + lo \cdot \frac{\frac{x}{hi} + -1}{hi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2}}{\frac{\left(x - lo\right) + lo \cdot \frac{\mathsf{fma}\left(-1, x, lo\right)}{hi}}{hi}}\\ \end{array} \]

Alternative 3: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{x}{hi}} \cdot \left(1 + lo \cdot \frac{\frac{x}{hi} + -1}{hi}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log (* (exp (/ x hi)) (+ 1.0 (* lo (/ (+ (/ x hi) -1.0) hi))))))

C

double code(double lo, double hi, double x) {
	return log((exp((x / hi)) * (1.0 + (lo * (((x / hi) + -1.0) / 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 = log((exp((x / hi)) * (1.0d0 + (lo * (((x / hi) + (-1.0d0)) / hi)))))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.log((Math.exp((x / hi)) * (1.0 + (lo * (((x / hi) + -1.0) / hi)))));
}

Python

def code(lo, hi, x):
	return math.log((math.exp((x / hi)) * (1.0 + (lo * (((x / hi) + -1.0) / hi)))))

Julia

function code(lo, hi, x)
	return log(Float64(exp(Float64(x / hi)) * Float64(1.0 + Float64(lo * Float64(Float64(Float64(x / hi) + -1.0) / hi)))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = log((exp((x / hi)) * (1.0 + (lo * (((x / hi) + -1.0) / hi)))));
end

Wolfram

code[lo_, hi_, x_] := N[Log[N[(N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(lo * N[(N[(N[(x / hi), $MachinePrecision] + -1.0), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. add-log-exp 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in lo around 0 20.6%

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

   1. associate-*r* 20.6%

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

   2. distribute-lft1-in 20.6%

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

9. Simplified 20.6%

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

10. Final simplification 20.6%

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

Alternative 4: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + \frac{x - lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log (+ 1.0 (/ (- x lo) hi))))

C

double code(double lo, double hi, double x) {
	return log((1.0 + ((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 = log((1.0d0 + ((x - lo) / hi)))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.log((1.0 + ((x - lo) / hi)));
}

Python

def code(lo, hi, x):
	return math.log((1.0 + ((x - lo) / hi)))

Julia

function code(lo, hi, x)
	return log(Float64(1.0 + Float64(Float64(x - lo) / hi)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = log((1.0 + ((x - lo) / hi)));
end

Wolfram

code[lo_, hi_, x_] := N[Log[N[(1.0 + N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. add-log-exp 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 20.6%

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

   1. +-commutative 20.6%

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

   2. associate--l+ 20.6%

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

   3. div-sub 20.6%

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

9. Simplified 20.6%

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

10. Final simplification 20.6%

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

Alternative 5: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{hi}{lo}\right| \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs((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 = abs((hi / lo))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.abs((hi / lo));
}

Python

def code(lo, hi, x):
	return math.fabs((hi / lo))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 9.5%

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

   1. +-commutative 9.5%

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

   2. associate--l+ 9.5%

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

   3. associate-*r/ 9.5%

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

   4. associate-*r/ 9.5%

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

   5. div-sub 9.5%

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

   6. distribute-lft-out-- 9.5%

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

   7. associate-*r/ 9.5%

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

   8. mul-1-neg 9.5%

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

   9. unsub-neg 9.5%

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

4. Simplified 9.5%

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

   1. add-sqr-sqrt 8.7%

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

   2. sqrt-unprod 18.1%

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

   3. pow2 18.1%

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

6. Applied egg-rr 18.1%

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

   1. unpow2 18.1%

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

   2. rem-sqrt-square 18.1%

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

   3. div-sub 18.1%

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

   4. associate-+l- 18.1%

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

   5. sub-neg 18.1%

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

   6. +-commutative 18.1%

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

   7. associate-+r+ 18.1%

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

   8. +-commutative 18.1%

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

   9. sub-neg 18.1%

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

   10. associate--l+ 18.1%

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

   11. div-sub 18.1%

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

8. Simplified 18.1%

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

9. Taylor expanded in hi around inf 19.3%

   \[\leadsto \left|\color{blue}{\frac{hi}{lo}}\right| \]

10. Final simplification 19.3%

    \[\leadsto \left|\frac{hi}{lo}\right| \]

Alternative 6: 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 7: 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 8: 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 2023240 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))