xlohi (overflows)

Percentage Accurate: 3.1% → 80.8%

Time: 31.5s

Alternatives: 7

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: 80.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{2}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot \frac{1}{1 + \left(\frac{x}{lo} + \frac{hi \cdot \frac{x}{lo} - hi}{lo}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - \frac{{x}^{2}}{{lo}^{2}}\right) \cdot \frac{1}{1 + \left(\frac{x}{lo} + hi \cdot \left(\frac{x}{{lo}^{2}} + \frac{-1}{lo}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{1 + \left(\frac{x}{lo} + hi \cdot \left(\frac{-1}{lo} - \frac{\frac{x}{lo}}{lo}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow (* (/ (- hi x) lo) (+ 1.0 (/ hi lo))) 2.0))))
   (if (<= x -1.35e+154)
     (* t_0 (/ 1.0 (+ 1.0 (+ (/ x lo) (/ (- (* hi (/ x lo)) hi) lo)))))
     (if (<= x 1.32e+154)
       (*
        (- 1.0 (/ (pow x 2.0) (pow lo 2.0)))
        (/ 1.0 (+ 1.0 (+ (/ x lo) (* hi (+ (/ x (pow lo 2.0)) (/ -1.0 lo)))))))
       (*
        t_0
        (/
         1.0
         (+ 1.0 (+ (/ x lo) (* hi (- (/ -1.0 lo) (/ (/ x lo) lo)))))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = 1.0 - pow((((hi - x) / lo) * (1.0 + (hi / lo))), 2.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (((hi * (x / lo)) - hi) / lo))));
	} else if (x <= 1.32e+154) {
		tmp = (1.0 - (pow(x, 2.0) / pow(lo, 2.0))) * (1.0 / (1.0 + ((x / lo) + (hi * ((x / pow(lo, 2.0)) + (-1.0 / lo))))));
	} else {
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (hi * ((-1.0 / lo) - ((x / lo) / lo))))));
	}
	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 = 1.0d0 - ((((hi - x) / lo) * (1.0d0 + (hi / lo))) ** 2.0d0)
    if (x <= (-1.35d+154)) then
        tmp = t_0 * (1.0d0 / (1.0d0 + ((x / lo) + (((hi * (x / lo)) - hi) / lo))))
    else if (x <= 1.32d+154) then
        tmp = (1.0d0 - ((x ** 2.0d0) / (lo ** 2.0d0))) * (1.0d0 / (1.0d0 + ((x / lo) + (hi * ((x / (lo ** 2.0d0)) + ((-1.0d0) / lo))))))
    else
        tmp = t_0 * (1.0d0 / (1.0d0 + ((x / lo) + (hi * (((-1.0d0) / lo) - ((x / lo) / lo))))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = 1.0 - Math.pow((((hi - x) / lo) * (1.0 + (hi / lo))), 2.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (((hi * (x / lo)) - hi) / lo))));
	} else if (x <= 1.32e+154) {
		tmp = (1.0 - (Math.pow(x, 2.0) / Math.pow(lo, 2.0))) * (1.0 / (1.0 + ((x / lo) + (hi * ((x / Math.pow(lo, 2.0)) + (-1.0 / lo))))));
	} else {
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (hi * ((-1.0 / lo) - ((x / lo) / lo))))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = 1.0 - math.pow((((hi - x) / lo) * (1.0 + (hi / lo))), 2.0)
	tmp = 0
	if x <= -1.35e+154:
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (((hi * (x / lo)) - hi) / lo))))
	elif x <= 1.32e+154:
		tmp = (1.0 - (math.pow(x, 2.0) / math.pow(lo, 2.0))) * (1.0 / (1.0 + ((x / lo) + (hi * ((x / math.pow(lo, 2.0)) + (-1.0 / lo))))))
	else:
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (hi * ((-1.0 / lo) - ((x / lo) / lo))))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(1.0 - (Float64(Float64(Float64(hi - x) / lo) * Float64(1.0 + Float64(hi / lo))) ^ 2.0))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(t_0 * Float64(1.0 / Float64(1.0 + Float64(Float64(x / lo) + Float64(Float64(Float64(hi * Float64(x / lo)) - hi) / lo)))));
	elseif (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 - Float64((x ^ 2.0) / (lo ^ 2.0))) * Float64(1.0 / Float64(1.0 + Float64(Float64(x / lo) + Float64(hi * Float64(Float64(x / (lo ^ 2.0)) + Float64(-1.0 / lo)))))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(1.0 + Float64(Float64(x / lo) + Float64(hi * Float64(Float64(-1.0 / lo) - Float64(Float64(x / lo) / lo)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = 1.0 - ((((hi - x) / lo) * (1.0 + (hi / lo))) ^ 2.0);
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (((hi * (x / lo)) - hi) / lo))));
	elseif (x <= 1.32e+154)
		tmp = (1.0 - ((x ^ 2.0) / (lo ^ 2.0))) * (1.0 / (1.0 + ((x / lo) + (hi * ((x / (lo ^ 2.0)) + (-1.0 / lo))))));
	else
		tmp = t_0 * (1.0 / (1.0 + ((x / lo) + (hi * ((-1.0 / lo) - ((x / lo) / lo))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 - N[Power[N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(t$95$0 * N[(1.0 / N[(1.0 + N[(N[(x / lo), $MachinePrecision] + N[(N[(N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(N[(1.0 - N[(N[Power[x, 2.0], $MachinePrecision] / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[(x / lo), $MachinePrecision] + N[(hi * N[(N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 + N[(N[(x / lo), $MachinePrecision] + N[(hi * N[(N[(-1.0 / lo), $MachinePrecision] - N[(N[(x / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 3.1%

      \[\frac{x - lo}{hi - lo} \]
   2. Add Preprocessing

   3. Taylor expanded in lo around inf 0.0%

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

   5. Simplified 18.8%

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

      1. flip-+ 18.8%

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

      2. div-inv 18.8%

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

      3. metadata-eval 18.8%

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

      4. pow2 18.8%

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

   7. Applied egg-rr 18.8%

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

   8. Taylor expanded in hi around 0 26.7%

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

   9. Taylor expanded in lo around inf 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

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

       3. associate-/l* 26.7%

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

   11. Simplified 26.7%

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

   if -1.35000000000000003e154 < x < 1.31999999999999998e154

   1. Initial program 3.1%

      \[\frac{x - lo}{hi - lo} \]
   2. Add Preprocessing

   3. Taylor expanded in lo around inf 0.0%

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

   5. Simplified 18.8%

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

      1. flip-+ 18.8%

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

      2. div-inv 18.8%

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

      3. metadata-eval 18.8%

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

      4. pow2 18.8%

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

   7. Applied egg-rr 18.8%

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

   8. Taylor expanded in hi around 0 25.9%

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

   9. Taylor expanded in hi around 0 99.0%

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

   if 1.31999999999999998e154 < x

   1. Initial program 3.1%

      \[\frac{x - lo}{hi - lo} \]
   2. Add Preprocessing

   3. Taylor expanded in lo around inf 0.0%

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

   5. Simplified 19.0%

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

      1. flip-+ 19.0%

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

      2. div-inv 19.0%

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

      3. metadata-eval 19.0%

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

      4. pow2 19.0%

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

   7. Applied egg-rr 19.0%

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

   8. Taylor expanded in hi around 0 27.7%

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

      1. add-sqr-sqrt 27.7%

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

      2. sqrt-unprod 27.7%

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

      3. sqr-neg 27.7%

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

      4. mul-1-neg 27.7%

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

      5. mul-1-neg 27.7%

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

      6. sqrt-unprod 27.7%

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

      7. add-sqr-sqrt 27.7%

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

      8. associate-*r/ 27.7%

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

      9. unpow2 27.7%

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

      10. times-frac 27.7%

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

   10. Applied egg-rr 27.7%

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

       1. associate-*l/ 27.7%

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

       2. mul-1-neg 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{2}\right) \cdot \frac{1}{1 + \left(\frac{x}{lo} + \frac{hi \cdot \frac{x}{lo} - hi}{lo}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - \frac{{x}^{2}}{{lo}^{2}}\right) \cdot \frac{1}{1 + \left(\frac{x}{lo} + hi \cdot \left(\frac{x}{{lo}^{2}} + \frac{-1}{lo}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{2}\right) \cdot \frac{1}{1 + \left(\frac{x}{lo} + hi \cdot \left(\frac{-1}{lo} - \frac{\frac{x}{lo}}{lo}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 26.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (1.0 - pow((((hi - x) / lo) * (1.0 + (hi / lo))), 2.0)) * (1.0 / (1.0 + ((x / lo) + (hi * ((-1.0 / lo) - ((x / lo) / 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 = (1.0d0 - ((((hi - x) / lo) * (1.0d0 + (hi / lo))) ** 2.0d0)) * (1.0d0 / (1.0d0 + ((x / lo) + (hi * (((-1.0d0) / lo) - ((x / lo) / lo))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

   1. flip-+ 18.9%

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

   2. div-inv 18.9%

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

   3. metadata-eval 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around 0 26.3%

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

   1. add-sqr-sqrt 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. sqr-neg 26.3%

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

   4. mul-1-neg 26.3%

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

   5. mul-1-neg 26.3%

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

   6. sqrt-unprod 26.3%

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

   7. add-sqr-sqrt 26.3%

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

   8. associate-*r/ 26.3%

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

   9. unpow2 26.3%

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

   10. times-frac 26.3%

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

10. Applied egg-rr 26.3%

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

    1. associate-*l/ 26.3%

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

    2. mul-1-neg 26.3%

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

12. Simplified 26.3%

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

13. Final simplification 26.3%

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

Alternative 3: 26.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ \left(1 + \left(t\_0 \cdot \left(1 + \frac{hi}{lo}\right)\right) \cdot \left(t\_0 \cdot \left(-1 - \frac{hi}{lo}\right)\right)\right) \cdot \frac{1}{1 + \frac{x - hi}{lo}} \end{array} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return (1.0 + ((t_0 * (1.0 + (hi / lo))) * (t_0 * (-1.0 - (hi / lo))))) * (1.0 / (1.0 + ((x - 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
    real(8) :: t_0
    t_0 = (hi - x) / lo
    code = (1.0d0 + ((t_0 * (1.0d0 + (hi / lo))) * (t_0 * ((-1.0d0) - (hi / lo))))) * (1.0d0 / (1.0d0 + ((x - hi) / lo)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

   1. flip-+ 18.9%

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

   2. div-inv 18.9%

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

   3. metadata-eval 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

   1. unpow2 18.9%

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

   2. +-commutative 18.9%

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

   3. +-commutative 18.9%

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

9. Applied egg-rr 18.9%

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

10. Taylor expanded in hi around 0 26.3%

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

11. Final simplification 26.3%

    \[\leadsto \left(1 + \left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right) \cdot \left(\frac{hi - x}{lo} \cdot \left(-1 - \frac{hi}{lo}\right)\right)\right) \cdot \frac{1}{1 + \frac{x - hi}{lo}} \]
12. Add Preprocessing

Alternative 4: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(lo, hi, x):
	return 1.0 + (hi * ((1.0 + (hi / lo)) / lo))

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
10. Add Preprocessing

Alternative 5: 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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

   \[\leadsto \frac{x - lo}{hi} \]
5. Add Preprocessing

Alternative 6: 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(lo / Float64(-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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
8. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))