xlohi (overflows)

Percentage Accurate: 3.1% → 21.8%

Time: 14.7s

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: 21.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathsf{expm1}\left(t\_0 + -0.5 \cdot {t\_0}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi))) (expm1 (+ t_0 (* -0.5 (pow t_0 2.0))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return expm1((t_0 + (-0.5 * pow(t_0, 2.0))));
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return Math.expm1((t_0 + (-0.5 * Math.pow(t_0, 2.0))));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return math.expm1((t_0 + (-0.5 * math.pow(t_0, 2.0))))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return expm1(Float64(t_0 + Float64(-0.5 * (t_0 ^ 2.0))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(Exp[N[(t$95$0 + N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $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. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right) - \frac{lo}{hi}}\right) \]
9. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)}\right) \]

   2. div-sub 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \color{blue}{\frac{x - lo}{hi}}\right) \]

   3. fma-define 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, \frac{x - lo}{hi}\right)}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, \frac{x - lo}{hi}\right)\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, \frac{x - lo}{hi}\right)\right) \]

   6. times-frac 21.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, \frac{x - lo}{hi}\right)\right) \]

   7. unpow2 21.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}, \frac{x - lo}{hi}\right)\right) \]

10. Simplified 21.8%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}\right)}\right) \]
11. Step-by-step derivation

    1. fma-undefine 21.8%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2} + \frac{x - lo}{hi}}\right) \]

    2. +-commutative 21.8%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{x - lo}{hi} + -0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2}}\right) \]

12. Applied egg-rr 21.8%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{x - lo}{hi} + -0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2}}\right) \]

13. Final simplification 21.8%

    \[\leadsto \mathsf{expm1}\left(\frac{x - lo}{hi} + -0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right) \]
14. Add Preprocessing

Alternative 2: 21.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\left(x - lo\right) \cdot \frac{{hi}^{-0.5}}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (expm1
  (fma
   -0.5
   (pow (* (- x lo) (/ (pow hi -0.5) (sqrt hi))) 2.0)
   (/ (- x lo) hi))))

C

double code(double lo, double hi, double x) {
	return expm1(fma(-0.5, pow(((x - lo) * (pow(hi, -0.5) / sqrt(hi))), 2.0), ((x - lo) / hi)));
}

Julia

function code(lo, hi, x)
	return expm1(fma(-0.5, (Float64(Float64(x - lo) * Float64((hi ^ -0.5) / sqrt(hi))) ^ 2.0), Float64(Float64(x - lo) / hi)))
end

Wolfram

code[lo_, hi_, x_] := N[(Exp[N[(-0.5 * N[Power[N[(N[(x - lo), $MachinePrecision] * N[(N[Power[hi, -0.5], $MachinePrecision] / N[Sqrt[hi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]] - 1), $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. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right) - \frac{lo}{hi}}\right) \]
9. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)}\right) \]

   2. div-sub 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \color{blue}{\frac{x - lo}{hi}}\right) \]

   3. fma-define 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, \frac{x - lo}{hi}\right)}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, \frac{x - lo}{hi}\right)\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, \frac{x - lo}{hi}\right)\right) \]

   6. times-frac 21.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, \frac{x - lo}{hi}\right)\right) \]

   7. unpow2 21.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}, \frac{x - lo}{hi}\right)\right) \]

10. Simplified 21.8%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}\right)}\right) \]
11. Step-by-step derivation

    1. *-un-lft-identity 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\frac{\color{blue}{1 \cdot \left(x - lo\right)}}{hi}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    2. add-sqr-sqrt 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\frac{1 \cdot \left(x - lo\right)}{\color{blue}{\sqrt{hi} \cdot \sqrt{hi}}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    3. times-frac 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\color{blue}{\left(\frac{1}{\sqrt{hi}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}}^{2}, \frac{x - lo}{hi}\right)\right) \]

    4. metadata-eval 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{hi}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    5. sqrt-div 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\color{blue}{\sqrt{\frac{1}{hi}}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    6. inv-pow 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\sqrt{\color{blue}{{hi}^{-1}}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    7. sqrt-pow1 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\color{blue}{{hi}^{\left(\frac{-1}{2}\right)}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    8. metadata-eval 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left({hi}^{\color{blue}{-0.5}} \cdot \frac{x - lo}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

12. Applied egg-rr 21.8%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\color{blue}{\left({hi}^{-0.5} \cdot \frac{x - lo}{\sqrt{hi}}\right)}}^{2}, \frac{x - lo}{hi}\right)\right) \]
13. Step-by-step derivation

    1. associate-*r/ 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\color{blue}{\left(\frac{{hi}^{-0.5} \cdot \left(x - lo\right)}{\sqrt{hi}}\right)}}^{2}, \frac{x - lo}{hi}\right)\right) \]

    2. *-commutative 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\frac{\color{blue}{\left(x - lo\right) \cdot {hi}^{-0.5}}}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]

    3. associate-/l* 21.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\color{blue}{\left(\left(x - lo\right) \cdot \frac{{hi}^{-0.5}}{\sqrt{hi}}\right)}}^{2}, \frac{x - lo}{hi}\right)\right) \]

14. Simplified 21.8%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\color{blue}{\left(\left(x - lo\right) \cdot \frac{{hi}^{-0.5}}{\sqrt{hi}}\right)}}^{2}, \frac{x - lo}{hi}\right)\right) \]

15. Final simplification 21.8%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\left(x - lo\right) \cdot \frac{{hi}^{-0.5}}{\sqrt{hi}}\right)}^{2}, \frac{x - lo}{hi}\right)\right) \]
16. Add Preprocessing

Alternative 3: 20.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow3 18.8%

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

5. Applied egg-rr 18.8%

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

   1. rem-cbrt-cube 18.8%

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

   2. expm1-log1p-u 18.8%

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

   3. expm1-undefine 18.8%

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

   4. flip-- 18.8%

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

   5. log1p-undefine 18.8%

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

   6. rem-exp-log 18.8%

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

   7. +-commutative 18.8%

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

   8. log1p-undefine 18.8%

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

   9. rem-exp-log 18.8%

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

   10. +-commutative 18.8%

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

   11. metadata-eval 18.8%

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

   12. log1p-undefine 18.8%

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

   13. rem-exp-log 18.8%

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

   14. +-commutative 18.8%

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

7. Applied egg-rr 18.8%

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

   1. difference-of-sqr-1 18.8%

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

   2. +-commutative 18.8%

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

   3. +-commutative 18.8%

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

   4. associate-+r+ 18.8%

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

   5. metadata-eval 18.8%

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

   6. associate--l+ 18.8%

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

   7. metadata-eval 18.8%

      \[\leadsto \frac{\left(2 + \frac{x - lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} + \color{blue}{0}\right)}{\left(\frac{x - lo}{hi} + 1\right) + 1} \]

   8. +-rgt-identity 18.8%

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

   9. +-commutative 18.8%

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

   10. +-commutative 18.8%

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

   11. associate-+r+ 18.8%

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

   12. metadata-eval 18.8%

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

9. Simplified 18.8%

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

10. Taylor expanded in lo around 0 20.9%

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

11. Final simplification 20.9%

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

Alternative 4: 20.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = (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[(2.0 * t$95$0), $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. add-cbrt-cube 18.8%

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

   2. pow3 18.8%

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

5. Applied egg-rr 18.8%

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

   1. rem-cbrt-cube 18.8%

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

   2. expm1-log1p-u 18.8%

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

   3. expm1-undefine 18.8%

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

   4. flip-- 18.8%

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

   5. log1p-undefine 18.8%

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

   6. rem-exp-log 18.8%

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

   7. +-commutative 18.8%

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

   8. log1p-undefine 18.8%

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

   9. rem-exp-log 18.8%

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

   10. +-commutative 18.8%

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

   11. metadata-eval 18.8%

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

   12. log1p-undefine 18.8%

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

   13. rem-exp-log 18.8%

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

   14. +-commutative 18.8%

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

7. Applied egg-rr 18.8%

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

   1. difference-of-sqr-1 18.8%

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

   2. +-commutative 18.8%

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

   3. +-commutative 18.8%

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

   4. associate-+r+ 18.8%

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

   5. metadata-eval 18.8%

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

   6. associate--l+ 18.8%

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

   7. metadata-eval 18.8%

      \[\leadsto \frac{\left(2 + \frac{x - lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} + \color{blue}{0}\right)}{\left(\frac{x - lo}{hi} + 1\right) + 1} \]

   8. +-rgt-identity 18.8%

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

   9. +-commutative 18.8%

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

   10. +-commutative 18.8%

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

   11. associate-+r+ 18.8%

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

   12. metadata-eval 18.8%

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

9. Simplified 18.8%

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

10. Taylor expanded in hi around inf 20.9%

    \[\leadsto \frac{\color{blue}{2 \cdot \frac{x - lo}{hi}}}{2 + \frac{x - lo}{hi}} \]

11. Final simplification 20.9%

    \[\leadsto \frac{2 \cdot \frac{x - lo}{hi}}{2 + \frac{x - lo}{hi}} \]
12. 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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
10. 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 18.6%

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

4. Final simplification 18.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))