xlohi (overflows)

Percentage Accurate: 3.1% → 22.2%

Time: 13.0s

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: 22.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{hi}\\ \sqrt[3]{\mathsf{fma}\left(lo, \frac{{t\_0}^{2}}{hi} \cdot -3, {t\_0}^{3}\right)} + -1 \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied egg-rr 18.8%

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

   1. add-cbrt-cube 18.8%

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

   2. pow3 18.8%

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

   3. log1p-undefine 18.8%

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

   4. rem-exp-log 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around 0 22.2%

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

   1. fma-define 22.2%

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

   2. distribute-rgt-out 22.2%

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

   3. metadata-eval 22.2%

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

10. Simplified 22.2%

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

11. Final simplification 22.2%

    \[\leadsto \sqrt[3]{\mathsf{fma}\left(lo, \frac{{\left(1 + \frac{x}{hi}\right)}^{2}}{hi} \cdot -3, {\left(1 + \frac{x}{hi}\right)}^{3}\right)} + -1 \]
12. Add Preprocessing

Alternative 2: 22.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{1 + \left(\frac{x}{hi} \cdot 3 - 3 \cdot \frac{lo}{hi}\right)} + -1 \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ (cbrt (+ 1.0 (- (* (/ x hi) 3.0) (* 3.0 (/ lo hi))))) -1.0))

C

double code(double lo, double hi, double x) {
	return cbrt((1.0 + (((x / hi) * 3.0) - (3.0 * (lo / hi))))) + -1.0;
}

Java

public static double code(double lo, double hi, double x) {
	return Math.cbrt((1.0 + (((x / hi) * 3.0) - (3.0 * (lo / hi))))) + -1.0;
}

Julia

function code(lo, hi, x)
	return Float64(cbrt(Float64(1.0 + Float64(Float64(Float64(x / hi) * 3.0) - Float64(3.0 * Float64(lo / hi))))) + -1.0)
end

Wolfram

code[lo_, hi_, x_] := N[(N[Power[N[(1.0 + N[(N[(N[(x / hi), $MachinePrecision] * 3.0), $MachinePrecision] - N[(3.0 * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $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. 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-undefine 18.8%

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

5. Applied egg-rr 18.8%

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

   1. add-cbrt-cube 18.8%

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

   2. pow3 18.8%

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

   3. log1p-undefine 18.8%

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

   4. rem-exp-log 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around inf 22.2%

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

   1. associate--l+ 22.2%

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

   2. distribute-lft1-in 22.2%

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

   3. metadata-eval 22.2%

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

   4. distribute-lft1-in 22.2%

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

   5. metadata-eval 22.2%

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

10. Simplified 22.2%

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

11. Final simplification 22.2%

    \[\leadsto \sqrt[3]{1 + \left(\frac{x}{hi} \cdot 3 - 3 \cdot \frac{lo}{hi}\right)} + -1 \]
12. Add Preprocessing

Alternative 3: 20.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{1}{2 + t\_0} \cdot \left(2 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (1.0 / (2.0 + t_0)) * (2.0 * 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)) * (2.0d0 * 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)) * (2.0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $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. 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-undefine 18.8%

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

5. Applied egg-rr 18.8%

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

   1. expm1-define 18.8%

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

7. Simplified 18.8%

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

   1. expm1-define 18.8%

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

   2. flip-- 18.8%

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

   3. div-inv 18.8%

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

   4. metadata-eval 18.8%

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

   5. sub-neg 18.8%

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

   6. pow2 18.8%

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

   7. log1p-undefine 18.8%

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

   8. rem-exp-log 18.8%

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

   9. metadata-eval 18.8%

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

   10. +-commutative 18.8%

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

   11. log1p-undefine 18.8%

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

   12. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

    1. *-commutative 18.8%

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

    2. associate-+r+ 18.8%

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

    3. metadata-eval 18.8%

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

    4. +-commutative 18.8%

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

11. Simplified 18.8%

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

12. Taylor expanded in hi around inf 3.1%

    \[\leadsto \frac{1}{2 + \frac{x - lo}{hi}} \cdot \color{blue}{\frac{2 \cdot x - 2 \cdot lo}{hi}} \]
13. Step-by-step derivation

    1. distribute-lft-out-- 3.1%

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

    2. associate-*r/ 20.9%

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

14. Simplified 20.9%

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

15. Final simplification 20.9%

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

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(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. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 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. 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 2024040 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))