xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 18.0s

Alternatives: 8

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: 18.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around -inf 10.4%

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

   1. mul-1-neg 10.4%

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

   2. distribute-neg-frac2 10.4%

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

   3. +-commutative 10.4%

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

   4. associate-/l* 18.9%

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

   5. *-commutative 18.9%

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

   6. distribute-lft-out 18.9%

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

6. Simplified 18.9%

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

   1. expm1-log1p-u 18.9%

      \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   2. expm1-undefine 18.9%

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

8. Applied egg-rr 18.9%

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

   1. expm1-define 18.9%

      \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

10. Simplified 18.9%

    \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 18.9%

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

    2. rem-exp-log 18.9%

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

    3. add-cube-cbrt 18.9%

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

    4. exp-prod 18.9%

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

    5. cbrt-unprod 18.9%

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

    6. pow2 18.9%

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

12. Applied egg-rr 18.9%

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

13. Final simplification 18.9%

    \[\leadsto \frac{lo - x}{lo} - hi \cdot \left(\frac{x}{{lo}^{2}} + \left(\frac{-1}{lo} + \frac{\frac{hi \cdot \left(1 - \frac{x}{lo}\right)}{{\left(e^{\sqrt[3]{{\log \left(-lo\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(-lo\right)}\right)}}}{lo}\right)\right) \]
14. Add Preprocessing

Alternative 2: 18.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

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 = sqrt(log(-lo))
    code = ((lo - x) / lo) + (hi * (((1.0d0 / lo) + (((hi * ((-1.0d0) + (x / lo))) / (exp(t_0) ** t_0)) / lo)) - (x / (lo ** 2.0d0))))
end function

Java

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

Python

def code(lo, hi, x):
	t_0 = math.sqrt(math.log(-lo))
	return ((lo - x) / lo) + (hi * (((1.0 / lo) + (((hi * (-1.0 + (x / lo))) / math.pow(math.exp(t_0), t_0)) / lo)) - (x / math.pow(lo, 2.0))))

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Sqrt[N[Log[(-lo)], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi * N[(-1.0 + N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[t$95$0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around -inf 10.4%

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

   1. mul-1-neg 10.4%

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

   2. distribute-neg-frac2 10.4%

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

   3. +-commutative 10.4%

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

   4. associate-/l* 18.9%

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

   5. *-commutative 18.9%

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

   6. distribute-lft-out 18.9%

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

6. Simplified 18.9%

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

   1. expm1-log1p-u 18.9%

      \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   2. expm1-undefine 18.9%

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

8. Applied egg-rr 18.9%

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

   1. expm1-define 18.9%

      \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

10. Simplified 18.9%

    \[\leadsto -1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right)\right)}}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 18.9%

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

    2. rem-exp-log 18.9%

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

    3. add-sqr-sqrt 18.9%

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

    4. exp-prod 18.9%

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

12. Applied egg-rr 18.9%

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

13. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around -inf 10.4%

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

   1. mul-1-neg 10.4%

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

   2. distribute-neg-frac2 10.4%

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

   3. +-commutative 10.4%

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

   4. associate-/l* 18.9%

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

   5. *-commutative 18.9%

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

   6. distribute-lft-out 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

   1. mul-1-neg 18.9%

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

   2. unsub-neg 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

5. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] - N[(hi * N[(N[(-1.0 + N[(N[(x - hi), $MachinePrecision] / 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 hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} - hi \cdot \frac{-1 + \frac{x - hi}{lo}}{lo} \]
8. Add Preprocessing

Alternative 6: 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. Add Preprocessing

Alternative 7: 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%

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

   1. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 8: 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. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))