xlohi (overflows)

Percentage Accurate: 3.1% → 19.8%

Time: 14.8s

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: 19.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{hi} - \frac{x}{hi \cdot hi}\\ t_1 := \log t_0 - \log \left(\frac{-1}{lo}\right)\\ \mathsf{expm1}\left(\sqrt[3]{{t_1}^{3} - \frac{3 \cdot \left(\left(\frac{1}{t_0} + \frac{x}{hi \cdot t_0}\right) \cdot {t_1}^{2}\right)}{lo}}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 hi) (/ x (* hi hi))))
        (t_1 (- (log t_0) (log (/ -1.0 lo)))))
   (expm1
    (cbrt
     (-
      (pow t_1 3.0)
      (/ (* 3.0 (* (+ (/ 1.0 t_0) (/ x (* hi t_0))) (pow t_1 2.0))) lo))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (1.0 / hi) - (x / (hi * hi));
	double t_1 = log(t_0) - log((-1.0 / lo));
	return expm1(cbrt((pow(t_1, 3.0) - ((3.0 * (((1.0 / t_0) + (x / (hi * t_0))) * pow(t_1, 2.0))) / lo))));
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (1.0 / hi) - (x / (hi * hi));
	double t_1 = Math.log(t_0) - Math.log((-1.0 / lo));
	return Math.expm1(Math.cbrt((Math.pow(t_1, 3.0) - ((3.0 * (((1.0 / t_0) + (x / (hi * t_0))) * Math.pow(t_1, 2.0))) / lo))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(1.0 / hi) - Float64(x / Float64(hi * hi)))
	t_1 = Float64(log(t_0) - log(Float64(-1.0 / lo)))
	return expm1(cbrt(Float64((t_1 ^ 3.0) - Float64(Float64(3.0 * Float64(Float64(Float64(1.0 / t_0) + Float64(x / Float64(hi * t_0))) * (t_1 ^ 2.0))) / lo))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(1.0 / hi), $MachinePrecision] - N[(x / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[t$95$0], $MachinePrecision] - N[Log[N[(-1.0 / lo), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(Exp[N[Power[N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[(N[(3.0 * N[(N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(x / N[(hi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]] - 1), $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. expm1-log1p-u 18.8%

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

   2. associate-/r* 18.9%

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

   3. sub-div 18.9%

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

6. Applied egg-rr 18.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)} \]
7. Step-by-step derivation

   1. add-cbrt-cube 18.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right) \cdot \mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right) \cdot \mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)}}\right) \]

   2. pow3 18.9%

      \[\leadsto \mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)}^{3}}}\right) \]

   3. associate-*r/ 18.9%

      \[\leadsto \mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(\frac{x}{hi} - \color{blue}{\frac{lo \cdot \left(1 - \frac{x}{hi}\right)}{hi}}\right)\right)}^{3}}\right) \]

   4. sub-div 18.9%

      \[\leadsto \mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(\color{blue}{\frac{x - lo \cdot \left(1 - \frac{x}{hi}\right)}{hi}}\right)\right)}^{3}}\right) \]

8. Applied egg-rr 18.9%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(\frac{x - lo \cdot \left(1 - \frac{x}{hi}\right)}{hi}\right)\right)}^{3}}}\right) \]

9. Taylor expanded in lo around -inf 19.8%

   \[\leadsto \mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{3} + -1 \cdot \frac{2 \cdot \left(\left(\frac{1}{\frac{1}{hi} - \frac{x}{{hi}^{2}}} + \frac{x}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right) \cdot hi}\right) \cdot {\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{2}\right) + \left(\frac{1}{\frac{1}{hi} - \frac{x}{{hi}^{2}}} + \frac{x}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right) \cdot hi}\right) \cdot {\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{2}}{lo}}}\right) \]
10. Step-by-step derivation

    1. mul-1-neg 19.8%

       \[\leadsto \mathsf{expm1}\left(\sqrt[3]{{\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{3} + \color{blue}{\left(-\frac{2 \cdot \left(\left(\frac{1}{\frac{1}{hi} - \frac{x}{{hi}^{2}}} + \frac{x}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right) \cdot hi}\right) \cdot {\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{2}\right) + \left(\frac{1}{\frac{1}{hi} - \frac{x}{{hi}^{2}}} + \frac{x}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right) \cdot hi}\right) \cdot {\left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \log \left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)\right)}^{2}}{lo}\right)}}\right) \]

11. Simplified 19.8%

    \[\leadsto \mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(\log \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) - \log \left(\frac{-1}{lo}\right)\right)}^{3} - \frac{3 \cdot \left(\left(\frac{1}{\frac{1}{hi} - \frac{x}{hi \cdot hi}} + \frac{x}{hi \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)}\right) \cdot {\left(\log \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) - \log \left(\frac{-1}{lo}\right)\right)}^{2}\right)}{lo}}}\right) \]

12. Final simplification 19.8%

    \[\leadsto \mathsf{expm1}\left(\sqrt[3]{{\left(\log \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) - \log \left(\frac{-1}{lo}\right)\right)}^{3} - \frac{3 \cdot \left(\left(\frac{1}{\frac{1}{hi} - \frac{x}{hi \cdot hi}} + \frac{x}{hi \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)}\right) \cdot {\left(\log \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) - \log \left(\frac{-1}{lo}\right)\right)}^{2}\right)}{lo}}\right) \]

Alternative 2: 18.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[(lo * N[Exp[N[Log[N[(N[(1.0 - N[(x / hi), $MachinePrecision]), $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. div-inv 18.8%

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

   2. metadata-eval 18.8%

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

   3. frac-times 18.8%

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

   4. cancel-sign-sub-inv 18.8%

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

   5. add-sqr-sqrt 8.8%

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

   6. sqrt-unprod 14.8%

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

   7. sqr-neg 14.8%

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

   8. sqrt-unprod 10.0%

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

   9. add-sqr-sqrt 18.8%

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

   10. frac-times 18.8%

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

   11. metadata-eval 18.8%

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

   12. div-inv 18.8%

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

   13. add-exp-log 18.8%

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

   14. div-inv 18.8%

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

   15. metadata-eval 18.8%

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

   16. frac-times 18.8%

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

6. Applied egg-rr 18.9%

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

7. Final simplification 18.9%

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

Alternative 3: 18.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. expm1-log1p-u 18.8%

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

   2. associate-/r* 18.9%

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

   3. sub-div 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in lo around -inf 18.9%

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

   1. mul-1-neg 18.9%

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

   2. associate-*l/ 18.9%

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

   3. distribute-lft-neg-out 18.9%

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

   4. cancel-sign-sub-inv 18.9%

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

   5. *-commutative 18.9%

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

   6. associate-*r/ 18.9%

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

   7. div-sub 18.9%

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

   8. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

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

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 2023274 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))