xlohi (overflows)

Percentage Accurate: 3.1% → 20.1%

Time: 30.8s

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: 20.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left({\left(\frac{x - hi}{lo}\right)}^{2} \cdot 0.3333333333333333 + \frac{hi - x}{lo}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ 1.0 (+ (* (pow (/ (- x hi) lo) 2.0) 0.3333333333333333) (/ (- hi x) lo))))

C

double code(double lo, double hi, double x) {
	return 1.0 + ((pow(((x - hi) / lo), 2.0) * 0.3333333333333333) + ((hi - x) / 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 + (((((x - hi) / lo) ** 2.0d0) * 0.3333333333333333d0) + ((hi - x) / lo))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + ((Math.pow(((x - hi) / lo), 2.0) * 0.3333333333333333) + ((hi - x) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + ((math.pow(((x - hi) / lo), 2.0) * 0.3333333333333333) + ((hi - x) / lo))

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64((Float64(Float64(x - hi) / lo) ^ 2.0) * 0.3333333333333333) + Float64(Float64(hi - x) / lo)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (((((x - hi) / lo) ^ 2.0) * 0.3333333333333333) + ((hi - x) / lo));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[Power[N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[(hi - x), $MachinePrecision] / lo), $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-cube-cbrt 15.7%

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

   2. pow3 15.7%

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

7. Applied egg-rr 15.7%

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

8. Taylor expanded in lo around -inf 19.4%

   \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3} \]
9. Step-by-step derivation

   1. *-commutative 19.4%

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

10. Simplified 19.4%

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

11. Taylor expanded in lo around inf 0.0%

    \[\leadsto \color{blue}{1 + \left(-0.6666666666666666 \cdot \frac{x - hi}{lo} + \left(-0.3333333333333333 \cdot \frac{x - hi}{lo} + \left(0.1111111111111111 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}} + 0.2222222222222222 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}}\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto 1 + \color{blue}{\left(\left(-0.3333333333333333 \cdot \frac{x - hi}{lo} + \left(0.1111111111111111 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}} + 0.2222222222222222 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}}\right)\right) + -0.6666666666666666 \cdot \frac{x - hi}{lo}\right)} \]

    2. +-commutative 0.0%

       \[\leadsto 1 + \left(\color{blue}{\left(\left(0.1111111111111111 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}} + 0.2222222222222222 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}}\right) + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)} + -0.6666666666666666 \cdot \frac{x - hi}{lo}\right) \]

    3. *-commutative 0.0%

       \[\leadsto 1 + \left(\left(\left(0.1111111111111111 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}} + 0.2222222222222222 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}}\right) + \color{blue}{\frac{x - hi}{lo} \cdot -0.3333333333333333}\right) + -0.6666666666666666 \cdot \frac{x - hi}{lo}\right) \]

    4. associate-+l+ 0.0%

       \[\leadsto 1 + \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}} + 0.2222222222222222 \cdot \frac{{\left(x - hi\right)}^{2}}{{lo}^{2}}\right) + \left(\frac{x - hi}{lo} \cdot -0.3333333333333333 + -0.6666666666666666 \cdot \frac{x - hi}{lo}\right)\right)} \]

13. Simplified 20.1%

    \[\leadsto \color{blue}{1 + \left({\left(\frac{x - hi}{lo}\right)}^{2} \cdot 0.3333333333333333 + \frac{hi - x}{lo}\right)} \]
14. Add Preprocessing

Alternative 2: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \frac{1}{\frac{lo}{x - hi}} \cdot -0.3333333333333333\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (pow (+ 1.0 (* (/ 1.0 (/ lo (- x hi))) -0.3333333333333333)) 3.0))

C

double code(double lo, double hi, double x) {
	return pow((1.0 + ((1.0 / (lo / (x - hi))) * -0.3333333333333333)), 3.0);
}

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 + ((1.0d0 / (lo / (x - hi))) * (-0.3333333333333333d0))) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((1.0 + ((1.0 / (lo / (x - hi))) * -0.3333333333333333)), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow((1.0 + ((1.0 / (lo / (x - hi))) * -0.3333333333333333)), 3.0)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(1.0 / Float64(lo / Float64(x - hi))) * -0.3333333333333333)) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (1.0 + ((1.0 / (lo / (x - hi))) * -0.3333333333333333)) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(N[(1.0 / N[(lo / N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 3.0], $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-cube-cbrt 15.7%

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

   2. pow3 15.7%

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

7. Applied egg-rr 15.7%

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

8. Taylor expanded in lo around -inf 19.4%

   \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3} \]
9. Step-by-step derivation

   1. *-commutative 19.4%

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

10. Simplified 19.4%

    \[\leadsto {\color{blue}{\left(1 + \frac{x - hi}{lo} \cdot -0.3333333333333333\right)}}^{3} \]
11. Step-by-step derivation

    1. clear-num 19.4%

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

    2. inv-pow 19.4%

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

12. Applied egg-rr 19.4%

    \[\leadsto {\left(1 + \color{blue}{{\left(\frac{lo}{x - hi}\right)}^{-1}} \cdot -0.3333333333333333\right)}^{3} \]
13. Step-by-step derivation

    1. unpow-1 19.4%

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

14. Simplified 19.4%

    \[\leadsto {\left(1 + \color{blue}{\frac{1}{\frac{lo}{x - hi}}} \cdot -0.3333333333333333\right)}^{3} \]
15. Add Preprocessing

Alternative 3: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \frac{x - hi}{lo} \cdot -0.3333333333333333\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (pow (+ 1.0 (* (/ (- x hi) lo) -0.3333333333333333)) 3.0))

C

double code(double lo, double hi, double x) {
	return pow((1.0 + (((x - hi) / lo) * -0.3333333333333333)), 3.0);
}

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 + (((x - hi) / lo) * (-0.3333333333333333d0))) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((1.0 + (((x - hi) / lo) * -0.3333333333333333)), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow((1.0 + (((x - hi) / lo) * -0.3333333333333333)), 3.0)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(Float64(x - hi) / lo) * -0.3333333333333333)) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (1.0 + (((x - hi) / lo) * -0.3333333333333333)) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 3.0], $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-cube-cbrt 15.7%

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

   2. pow3 15.7%

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

7. Applied egg-rr 15.7%

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

8. Taylor expanded in lo around -inf 19.4%

   \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3} \]
9. Step-by-step derivation

   1. *-commutative 19.4%

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

10. Simplified 19.4%

    \[\leadsto {\color{blue}{\left(1 + \frac{x - hi}{lo} \cdot -0.3333333333333333\right)}}^{3} \]
11. Add Preprocessing

Alternative 4: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(1 + hi \cdot \frac{0.3333333333333333}{lo}\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (pow (+ 1.0 (* hi (/ 0.3333333333333333 lo))) 3.0))

C

double code(double lo, double hi, double x) {
	return pow((1.0 + (hi * (0.3333333333333333 / lo))), 3.0);
}

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 + (hi * (0.3333333333333333d0 / lo))) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((1.0 + (hi * (0.3333333333333333 / lo))), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow((1.0 + (hi * (0.3333333333333333 / lo))), 3.0)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(hi * Float64(0.3333333333333333 / lo))) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (1.0 + (hi * (0.3333333333333333 / lo))) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(hi * N[(0.3333333333333333 / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-cube-cbrt 15.7%

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

   2. pow3 15.7%

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

7. Applied egg-rr 15.7%

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

8. Taylor expanded in lo around -inf 19.4%

   \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3} \]
9. Step-by-step derivation

   1. *-commutative 19.4%

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

10. Simplified 19.4%

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

11. Taylor expanded in x around 0 19.4%

    \[\leadsto {\left(1 + \color{blue}{0.3333333333333333 \cdot \frac{hi}{lo}}\right)}^{3} \]
12. Step-by-step derivation

    1. *-commutative 19.4%

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

    2. *-rgt-identity 19.4%

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

    3. associate-*r/ 19.4%

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

    4. associate-*r* 19.4%

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

    5. *-commutative 19.4%

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

    6. associate-*r/ 19.4%

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

    7. metadata-eval 19.4%

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

13. Simplified 19.4%

    \[\leadsto {\left(1 + \color{blue}{hi \cdot \frac{0.3333333333333333}{lo}}\right)}^{3} \]
14. Add Preprocessing

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(1.0 + N[(N[(hi - x), $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 x around 0 19.0%

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

5. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

7. Simplified 19.0%

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

8. Final simplification 19.0%

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

Alternative 6: 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 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. Step-by-step derivation

   1. div-inv 18.9%

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

5. Applied egg-rr 18.9%

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

6. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

8. Simplified 18.9%

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

9. Taylor expanded in x around 0 19.0%

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

    1. +-commutative 19.0%

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

11. Simplified 19.0%

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

12. Final simplification 19.0%

    \[\leadsto 1 + \frac{hi \cdot \left(1 + \frac{hi}{lo}\right)}{lo} \]
13. 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(Float64(-lo) / hi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.7%

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

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. 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 2024172 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))