xlohi (overflows)

Percentage Accurate: 3.1% → 23.2%

Time: 9.0s

Alternatives: 5

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: 23.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.05275 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(1 + 2 \cdot t_0\right) + -1\right) \cdot \frac{1}{1 + \left(1 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {t_0}^{2}\right) \cdot \frac{1}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.05275e+308)
     (* (+ (+ 1.0 (* 2.0 t_0)) -1.0) (/ 1.0 (+ 1.0 (+ 1.0 t_0))))
     (*
      (- (pow (* (- x lo) (/ lo (* hi hi))) 2.0) (pow t_0 2.0))
      (/ 1.0 (/ (+ (* lo t_0) (- lo x)) hi))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.05275e+308) {
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	} else {
		tmp = (pow(((x - lo) * (lo / (hi * hi))), 2.0) - pow(t_0, 2.0)) * (1.0 / (((lo * t_0) + (lo - x)) / 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 (lo <= (-1.05275d+308)) then
        tmp = ((1.0d0 + (2.0d0 * t_0)) + (-1.0d0)) * (1.0d0 / (1.0d0 + (1.0d0 + t_0)))
    else
        tmp = ((((x - lo) * (lo / (hi * hi))) ** 2.0d0) - (t_0 ** 2.0d0)) * (1.0d0 / (((lo * t_0) + (lo - x)) / 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 (lo <= -1.05275e+308) {
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	} else {
		tmp = (Math.pow(((x - lo) * (lo / (hi * hi))), 2.0) - Math.pow(t_0, 2.0)) * (1.0 / (((lo * t_0) + (lo - x)) / hi));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.05275e+308:
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)))
	else:
		tmp = (math.pow(((x - lo) * (lo / (hi * hi))), 2.0) - math.pow(t_0, 2.0)) * (1.0 / (((lo * t_0) + (lo - x)) / hi))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.05275e+308)
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 * t_0)) + -1.0) * Float64(1.0 / Float64(1.0 + Float64(1.0 + t_0))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) * Float64(lo / Float64(hi * hi))) ^ 2.0) - (t_0 ^ 2.0)) * Float64(1.0 / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.05275e+308)
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	else
		tmp = ((((x - lo) * (lo / (hi * hi))) ^ 2.0) - (t_0 ^ 2.0)) * (1.0 / (((lo * t_0) + (lo - x)) / hi));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.05275e+308], N[(N[(N[(1.0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.05274999999999995e308

   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. Step-by-step derivation

      1. expm1-log1p-u 18.8%

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

      2. expm1-udef 18.8%

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

   4. Applied egg-rr 18.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x - lo}{hi}\right)} - 1} \]
   5. Step-by-step derivation

      1. flip-- 18.8%

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

      2. div-inv 18.8%

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

      3. metadata-eval 18.8%

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

      4. sub-neg 18.8%

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

      5. pow2 18.8%

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

      6. log1p-udef 18.8%

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

      7. add-exp-log 18.8%

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

      8. metadata-eval 18.8%

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

      9. log1p-udef 18.8%

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

      10. add-exp-log 18.8%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in hi around inf 20.9%

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

      1. associate--l+ 20.9%

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

      2. associate-*r/ 20.9%

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

      3. associate-*r/ 3.1%

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

      4. div-sub 3.1%

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

      5. distribute-lft-out-- 3.1%

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

      6. associate-*r/ 20.9%

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

   9. Simplified 20.9%

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

   if -1.05274999999999995e308 < 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 18.2%

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

      6. div-sub 18.2%

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

   4. Simplified 18.2%

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

      1. div-inv 18.2%

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

      2. inv-pow 18.2%

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

   6. Applied egg-rr 18.2%

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

      1. unpow-1 18.2%

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

   8. Simplified 18.2%

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

      1. flip-+ 18.2%

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

      2. div-inv 18.2%

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

      3. pow2 18.2%

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

      4. associate-*l* 18.2%

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

      5. frac-times 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. pow2 98.6%

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

   10. Applied egg-rr 51.2%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.05275 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(1 + 2 \cdot \frac{x - lo}{hi}\right) + -1\right) \cdot \frac{1}{1 + \left(1 + \frac{x - lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right) \cdot \frac{1}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 2: 23.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.05275 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(1 + 2 \cdot t_0\right) + -1\right) \cdot \frac{1}{1 + \left(1 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.05275e+308)
     (* (+ (+ 1.0 (* 2.0 t_0)) -1.0) (/ 1.0 (+ 1.0 (+ 1.0 t_0))))
     (/
      (- (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (pow t_0 2.0))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.05275e+308) {
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) - pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / 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 (lo <= (-1.05275d+308)) then
        tmp = ((1.0d0 + (2.0d0 * t_0)) + (-1.0d0)) * (1.0d0 / (1.0d0 + (1.0d0 + t_0)))
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) - (t_0 ** 2.0d0)) / (((lo * t_0) + (lo - x)) / 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 (lo <= -1.05275e+308) {
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - Math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.05275e+308:
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)))
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.05275e+308)
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 * t_0)) + -1.0) * Float64(1.0 / Float64(1.0 + Float64(1.0 + t_0))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.05275e+308)
		tmp = ((1.0 + (2.0 * t_0)) + -1.0) * (1.0 / (1.0 + (1.0 + t_0)));
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.05275e+308], N[(N[(N[(1.0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.05274999999999995e308

   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. Step-by-step derivation

      1. expm1-log1p-u 18.8%

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

      2. expm1-udef 18.8%

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

   4. Applied egg-rr 18.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x - lo}{hi}\right)} - 1} \]
   5. Step-by-step derivation

      1. flip-- 18.8%

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

      2. div-inv 18.8%

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

      3. metadata-eval 18.8%

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

      4. sub-neg 18.8%

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

      5. pow2 18.8%

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

      6. log1p-udef 18.8%

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

      7. add-exp-log 18.8%

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

      8. metadata-eval 18.8%

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

      9. log1p-udef 18.8%

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

      10. add-exp-log 18.8%

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

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in hi around inf 20.9%

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

      1. associate--l+ 20.9%

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

      2. associate-*r/ 20.9%

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

      3. associate-*r/ 3.1%

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

      4. div-sub 3.1%

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

      5. distribute-lft-out-- 3.1%

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

      6. associate-*r/ 20.9%

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

   9. Simplified 20.9%

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

   if -1.05274999999999995e308 < 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 18.2%

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

      6. div-sub 18.2%

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

   4. Simplified 18.2%

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

      1. flip-+ 18.2%

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

      2. pow2 18.2%

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

      3. *-commutative 18.2%

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

      4. clear-num 18.2%

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

      5. frac-times 58.0%

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

      6. *-un-lft-identity 58.0%

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

      7. pow2 58.0%

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

      8. associate-*r/ 58.0%

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

      9. sub-div 51.0%

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

   6. Applied egg-rr 51.0%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.05275 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(1 + 2 \cdot \frac{x - lo}{hi}\right) + -1\right) \cdot \frac{1}{1 + \left(1 + \frac{x - lo}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 3: 20.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. expm1-log1p-u 18.8%

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

   2. expm1-udef 18.8%

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

4. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x - lo}{hi}\right)} - 1} \]
5. Step-by-step derivation

   1. flip-- 18.8%

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

   2. div-inv 18.8%

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

   3. metadata-eval 18.8%

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

   4. sub-neg 18.8%

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

   5. pow2 18.8%

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

   6. log1p-udef 18.8%

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

   7. add-exp-log 18.8%

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

   8. metadata-eval 18.8%

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

   9. log1p-udef 18.8%

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

   10. add-exp-log 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 20.9%

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

   1. associate--l+ 20.9%

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

   2. associate-*r/ 20.9%

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

   3. associate-*r/ 3.1%

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

   4. div-sub 3.1%

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

   5. distribute-lft-out-- 3.1%

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

   6. associate-*r/ 20.9%

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

9. Simplified 20.9%

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

10. Final simplification 20.9%

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

Alternative 4: 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 5: 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 2023250 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))