xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 22.9s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{{lo}^{2}}\\ hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - t\_0\right)}{lo}\right) - t\_0\right) - \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{6}}{lo} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ x (pow lo 2.0))))
   (-
    (* hi (- (+ (/ 1.0 lo) (/ (* hi (- (/ 1.0 lo) t_0)) lo)) t_0))
    (/ (pow (pow (- x lo) 0.16666666666666666) 6.0) lo))))

C

double code(double lo, double hi, double x) {
	double t_0 = x / pow(lo, 2.0);
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (pow(pow((x - lo), 0.16666666666666666), 6.0) / lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x / (lo ** 2.0d0)
    code = (hi * (((1.0d0 / lo) + ((hi * ((1.0d0 / lo) - t_0)) / lo)) - t_0)) - ((((x - lo) ** 0.16666666666666666d0) ** 6.0d0) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = x / Math.pow(lo, 2.0);
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (Math.pow(Math.pow((x - lo), 0.16666666666666666), 6.0) / lo);
}

Python

def code(lo, hi, x):
	t_0 = x / math.pow(lo, 2.0)
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (math.pow(math.pow((x - lo), 0.16666666666666666), 6.0) / lo)

Julia

function code(lo, hi, x)
	t_0 = Float64(x / (lo ^ 2.0))
	return Float64(Float64(hi * Float64(Float64(Float64(1.0 / lo) + Float64(Float64(hi * Float64(Float64(1.0 / lo) - t_0)) / lo)) - t_0)) - Float64(((Float64(x - lo) ^ 0.16666666666666666) ^ 6.0) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = x / (lo ^ 2.0);
	tmp = (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - ((((x - lo) ^ 0.16666666666666666) ^ 6.0) / lo);
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Power[N[(x - lo), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 6.0], $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

   1. add-exp-log 19.0%

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

5. Applied egg-rr 19.0%

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

   1. rem-exp-log 19.0%

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

   2. rem-cube-cbrt 19.0%

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

   3. add-sqr-sqrt 19.0%

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

   4. unpow-prod-down 19.0%

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

   5. pow1/3 19.0%

      \[\leadsto -1 \cdot \frac{{\left(\sqrt{\color{blue}{{\left(x - lo\right)}^{0.3333333333333333}}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   6. sqrt-pow1 19.0%

      \[\leadsto -1 \cdot \frac{{\color{blue}{\left({\left(x - lo\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   7. metadata-eval 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   8. pow1/3 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(x - lo\right)}^{0.3333333333333333}}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   9. sqrt-pow1 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(x - lo\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   10. metadata-eval 19.0%

       \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(x - lo\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

7. Applied egg-rr 19.0%

   \[\leadsto -1 \cdot \frac{\color{blue}{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]
8. Step-by-step derivation

   1. pow-prod-up 19.0%

      \[\leadsto -1 \cdot \frac{\color{blue}{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{\left(3 + 3\right)}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   2. metadata-eval 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{\color{blue}{6}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

9. Applied egg-rr 19.0%

   \[\leadsto -1 \cdot \frac{\color{blue}{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{6}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

10. Final simplification 19.0%

    \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) - \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{6}}{lo} \]
11. Add Preprocessing

Alternative 2: 18.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{{lo}^{2}}\\ hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - t\_0\right)}{lo}\right) - t\_0\right) - \frac{e^{\log \left({\left(x - lo\right)}^{0.16666666666666666}\right) \cdot 6}}{lo} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ x (pow lo 2.0))))
   (-
    (* hi (- (+ (/ 1.0 lo) (/ (* hi (- (/ 1.0 lo) t_0)) lo)) t_0))
    (/ (exp (* (log (pow (- x lo) 0.16666666666666666)) 6.0)) lo))))

C

double code(double lo, double hi, double x) {
	double t_0 = x / pow(lo, 2.0);
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (exp((log(pow((x - lo), 0.16666666666666666)) * 6.0)) / lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x / (lo ** 2.0d0)
    code = (hi * (((1.0d0 / lo) + ((hi * ((1.0d0 / lo) - t_0)) / lo)) - t_0)) - (exp((log(((x - lo) ** 0.16666666666666666d0)) * 6.0d0)) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = x / Math.pow(lo, 2.0);
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (Math.exp((Math.log(Math.pow((x - lo), 0.16666666666666666)) * 6.0)) / lo);
}

Python

def code(lo, hi, x):
	t_0 = x / math.pow(lo, 2.0)
	return (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (math.exp((math.log(math.pow((x - lo), 0.16666666666666666)) * 6.0)) / lo)

Julia

function code(lo, hi, x)
	t_0 = Float64(x / (lo ^ 2.0))
	return Float64(Float64(hi * Float64(Float64(Float64(1.0 / lo) + Float64(Float64(hi * Float64(Float64(1.0 / lo) - t_0)) / lo)) - t_0)) - Float64(exp(Float64(log((Float64(x - lo) ^ 0.16666666666666666)) * 6.0)) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = x / (lo ^ 2.0);
	tmp = (hi * (((1.0 / lo) + ((hi * ((1.0 / lo) - t_0)) / lo)) - t_0)) - (exp((log(((x - lo) ^ 0.16666666666666666)) * 6.0)) / lo);
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[Log[N[Power[N[(x - lo), $MachinePrecision], 0.16666666666666666], $MachinePrecision]], $MachinePrecision] * 6.0), $MachinePrecision]], $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

   1. add-exp-log 19.0%

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

5. Applied egg-rr 19.0%

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

   1. rem-exp-log 19.0%

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

   2. rem-cube-cbrt 19.0%

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

   3. add-sqr-sqrt 19.0%

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

   4. unpow-prod-down 19.0%

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

   5. pow1/3 19.0%

      \[\leadsto -1 \cdot \frac{{\left(\sqrt{\color{blue}{{\left(x - lo\right)}^{0.3333333333333333}}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   6. sqrt-pow1 19.0%

      \[\leadsto -1 \cdot \frac{{\color{blue}{\left({\left(x - lo\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   7. metadata-eval 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{x - lo}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   8. pow1/3 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(x - lo\right)}^{0.3333333333333333}}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   9. sqrt-pow1 19.0%

      \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(x - lo\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   10. metadata-eval 19.0%

       \[\leadsto -1 \cdot \frac{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(x - lo\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

7. Applied egg-rr 19.0%

   \[\leadsto -1 \cdot \frac{\color{blue}{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{3}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]
8. Step-by-step derivation

   1. pow-prod-up 19.0%

      \[\leadsto -1 \cdot \frac{\color{blue}{{\left({\left(x - lo\right)}^{0.16666666666666666}\right)}^{\left(3 + 3\right)}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   2. pow-to-exp 19.0%

      \[\leadsto -1 \cdot \frac{\color{blue}{e^{\log \left({\left(x - lo\right)}^{0.16666666666666666}\right) \cdot \left(3 + 3\right)}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   3. metadata-eval 19.0%

      \[\leadsto -1 \cdot \frac{e^{\log \left({\left(x - lo\right)}^{0.16666666666666666}\right) \cdot \color{blue}{6}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

9. Applied egg-rr 19.0%

   \[\leadsto -1 \cdot \frac{\color{blue}{e^{\log \left({\left(x - lo\right)}^{0.16666666666666666}\right) \cdot 6}}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

10. Final simplification 19.0%

    \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) - \frac{e^{\log \left({\left(x - lo\right)}^{0.16666666666666666}\right) \cdot 6}}{lo} \]
11. Add Preprocessing

Alternative 3: 18.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((lo - x) / lo) - (hi * ((x / pow(lo, 2.0)) + ((-1.0 / lo) - exp(log(((hi * ((1.0 / lo) - (x * pow(lo, -2.0)))) / 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) / lo) - exp(log(((hi * ((1.0d0 / lo) - (x * (lo ** (-2.0d0))))) / 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 / lo) - Math.exp(Math.log(((hi * ((1.0 / lo) - (x * Math.pow(lo, -2.0)))) / lo))))));
}

Python

def code(lo, hi, x):
	return ((lo - x) / lo) - (hi * ((x / math.pow(lo, 2.0)) + ((-1.0 / lo) - math.exp(math.log(((hi * ((1.0 / lo) - (x * math.pow(lo, -2.0)))) / 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 / lo) - exp(log(Float64(Float64(hi * Float64(Float64(1.0 / lo) - Float64(x * (lo ^ -2.0)))) / lo)))))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((lo - x) / lo) - (hi * ((x / (lo ^ 2.0)) + ((-1.0 / lo) - exp(log(((hi * ((1.0 / lo) - (x * (lo ^ -2.0)))) / 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 / lo), $MachinePrecision] - N[Exp[N[Log[N[(N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - N[(x * N[Power[lo, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

   1. add-exp-log 19.0%

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

   2. div-inv 19.0%

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

   3. pow-flip 19.0%

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

   4. metadata-eval 19.0%

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

5. Applied egg-rr 19.0%

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

6. Final simplification 19.0%

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

Alternative 4: 18.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{\frac{lo}{\mathsf{fma}\left(hi, -1 - \frac{hi}{lo}, x\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 14.8%

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

   5. fma-define 14.8%

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

5. Simplified 14.8%

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

   1. clear-num 14.8%

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

   2. inv-pow 14.8%

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

7. Applied egg-rr 14.8%

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

   1. unpow-1 14.8%

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

   2. fma-define 14.8%

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

   3. associate-*r/ 3.1%

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

   4. +-commutative 3.1%

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

   5. associate--l+ 3.1%

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

   6. sub-neg 3.1%

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

   7. associate-*r/ 14.8%

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

   8. neg-mul-1 14.8%

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

   9. *-commutative 14.8%

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

   10. distribute-lft-in 18.9%

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

   11. div-sub 18.9%

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

   12. sub-neg 18.9%

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

   13. mul-1-neg 18.9%

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

   14. +-commutative 18.9%

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

   15. metadata-eval 18.9%

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

   16. sub-neg 18.9%

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

   17. +-commutative 18.9%

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

   18. fma-define 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in x around 0 19.0%

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

    1. mul-1-neg 19.0%

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

    2. distribute-neg-frac2 19.0%

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

12. Simplified 19.0%

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

13. Final simplification 19.0%

    \[\leadsto 1 + \frac{-1}{\frac{lo}{\mathsf{fma}\left(hi, -1 - \frac{hi}{lo}, x\right)}} \]
14. Add Preprocessing

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 + ((hi * (1.0d0 + (hi / lo))) / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 14.8%

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

   5. fma-define 14.8%

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

5. Simplified 14.8%

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

   1. clear-num 14.8%

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

   2. inv-pow 14.8%

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

7. Applied egg-rr 14.8%

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

   1. unpow-1 14.8%

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

   2. fma-define 14.8%

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

   3. associate-*r/ 3.1%

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

   4. +-commutative 3.1%

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

   5. associate--l+ 3.1%

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

   6. sub-neg 3.1%

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

   7. associate-*r/ 14.8%

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

   8. neg-mul-1 14.8%

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

   9. *-commutative 14.8%

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

   10. distribute-lft-in 18.9%

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

   11. div-sub 18.9%

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

   12. sub-neg 18.9%

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

   13. mul-1-neg 18.9%

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

   14. +-commutative 18.9%

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

   15. metadata-eval 18.9%

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

   16. sub-neg 18.9%

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

   17. +-commutative 18.9%

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

   18. fma-define 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in x around 0 18.9%

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

    1. mul-1-neg 18.9%

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

    \[\leadsto 1 + \frac{hi \cdot \left(1 + \frac{hi}{lo}\right)}{lo} \]
14. 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%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 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 2024114 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))