xlohi (overflows)

Percentage Accurate: 3.1% → 19.3%

Time: 11.6s

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: 19.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ {\left(\left(1 + \frac{x \cdot -0.3333333333333333}{lo}\right) + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (pow
  (+
   (+ 1.0 (/ (* x -0.3333333333333333) lo))
   (*
    0.3333333333333333
    (*
     (pow (/ 1.0 (pow (- 1.0 (/ x lo)) 2.0)) 0.1111111111111111)
     (* hi (- (/ 1.0 lo) (/ x (pow lo 2.0)))))))
  3.0))

C

double code(double lo, double hi, double x) {
	return pow(((1.0 + ((x * -0.3333333333333333) / lo)) + (0.3333333333333333 * (pow((1.0 / pow((1.0 - (x / lo)), 2.0)), 0.1111111111111111) * (hi * ((1.0 / lo) - (x / pow(lo, 2.0))))))), 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 * (-0.3333333333333333d0)) / lo)) + (0.3333333333333333d0 * (((1.0d0 / ((1.0d0 - (x / lo)) ** 2.0d0)) ** 0.1111111111111111d0) * (hi * ((1.0d0 / lo) - (x / (lo ** 2.0d0))))))) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow(((1.0 + ((x * -0.3333333333333333) / lo)) + (0.3333333333333333 * (Math.pow((1.0 / Math.pow((1.0 - (x / lo)), 2.0)), 0.1111111111111111) * (hi * ((1.0 / lo) - (x / Math.pow(lo, 2.0))))))), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow(((1.0 + ((x * -0.3333333333333333) / lo)) + (0.3333333333333333 * (math.pow((1.0 / math.pow((1.0 - (x / lo)), 2.0)), 0.1111111111111111) * (hi * ((1.0 / lo) - (x / math.pow(lo, 2.0))))))), 3.0)

Julia

function code(lo, hi, x)
	return Float64(Float64(1.0 + Float64(Float64(x * -0.3333333333333333) / lo)) + Float64(0.3333333333333333 * Float64((Float64(1.0 / (Float64(1.0 - Float64(x / lo)) ^ 2.0)) ^ 0.1111111111111111) * Float64(hi * Float64(Float64(1.0 / lo) - Float64(x / (lo ^ 2.0))))))) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((1.0 + ((x * -0.3333333333333333) / lo)) + (0.3333333333333333 * (((1.0 / ((1.0 - (x / lo)) ^ 2.0)) ^ 0.1111111111111111) * (hi * ((1.0 / lo) - (x / (lo ^ 2.0))))))) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(N[(1.0 + N[(N[(x * -0.3333333333333333), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[Power[N[(1.0 / N[Power[N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.1111111111111111], $MachinePrecision] * N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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* 15.6%

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

   5. fma-define 15.6%

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

5. Simplified 15.6%

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

   1. add-cube-cbrt 15.6%

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

   2. pow3 15.6%

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

7. Applied egg-rr 15.6%

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

8. Taylor expanded in hi around 0 19.3%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{1 - \frac{x}{lo}} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}}^{3} \]

9. Taylor expanded in x around 0 19.3%

   \[\leadsto {\left(\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x}{lo}\right)} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}^{3} \]
10. Step-by-step derivation

    1. associate-*r/ 19.3%

       \[\leadsto {\left(\left(1 + \color{blue}{\frac{-0.3333333333333333 \cdot x}{lo}}\right) + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}^{3} \]

    2. *-commutative 19.3%

       \[\leadsto {\left(\left(1 + \frac{\color{blue}{x \cdot -0.3333333333333333}}{lo}\right) + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}^{3} \]

11. Simplified 19.3%

    \[\leadsto {\left(\color{blue}{\left(1 + \frac{x \cdot -0.3333333333333333}{lo}\right)} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}^{3} \]
12. Add Preprocessing

Alternative 2: 19.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{lo}\\ {\left(\sqrt[3]{t\_0} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{t\_0}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \frac{1}{lo}\right)\right)\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x lo))))
   (pow
    (+
     (cbrt t_0)
     (*
      0.3333333333333333
      (* (pow (/ 1.0 (pow t_0 2.0)) 0.1111111111111111) (* hi (/ 1.0 lo)))))
    3.0)))

C

double code(double lo, double hi, double x) {
	double t_0 = 1.0 - (x / lo);
	return pow((cbrt(t_0) + (0.3333333333333333 * (pow((1.0 / pow(t_0, 2.0)), 0.1111111111111111) * (hi * (1.0 / lo))))), 3.0);
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = 1.0 - (x / lo);
	return Math.pow((Math.cbrt(t_0) + (0.3333333333333333 * (Math.pow((1.0 / Math.pow(t_0, 2.0)), 0.1111111111111111) * (hi * (1.0 / lo))))), 3.0);
}

Julia

function code(lo, hi, x)
	t_0 = Float64(1.0 - Float64(x / lo))
	return Float64(cbrt(t_0) + Float64(0.3333333333333333 * Float64((Float64(1.0 / (t_0 ^ 2.0)) ^ 0.1111111111111111) * Float64(hi * Float64(1.0 / lo))))) ^ 3.0
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]}, N[Power[N[(N[Power[t$95$0, 1/3], $MachinePrecision] + N[(0.3333333333333333 * N[(N[Power[N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 0.1111111111111111], $MachinePrecision] * N[(hi * N[(1.0 / lo), $MachinePrecision]), $MachinePrecision]), $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* 15.6%

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

   5. fma-define 15.6%

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

5. Simplified 15.6%

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

   1. add-cube-cbrt 15.6%

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

   2. pow3 15.6%

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

7. Applied egg-rr 15.6%

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

8. Taylor expanded in hi around 0 19.3%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{1 - \frac{x}{lo}} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}}^{3} \]

9. Taylor expanded in lo around inf 19.3%

   \[\leadsto {\left(\sqrt[3]{1 - \frac{x}{lo}} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \color{blue}{\frac{1}{lo}}\right)\right)\right)}^{3} \]
10. Add Preprocessing

Alternative 3: 19.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow((1.0 + (0.3333333333333333 * ((hi - x) / 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 + (0.3333333333333333d0 * ((hi - x) / lo))) ** 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(0.3333333333333333 * N[(N[(hi - x), $MachinePrecision] / 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* 15.6%

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

   5. fma-define 15.6%

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

5. Simplified 15.6%

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

   1. add-cube-cbrt 15.6%

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

   2. pow3 15.6%

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

7. Applied egg-rr 15.6%

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

8. Taylor expanded in lo around inf 19.3%

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

   1. *-commutative 19.3%

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

10. Simplified 19.3%

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

11. Final simplification 19.3%

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

Alternative 4: 19.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow((1.0 + (0.3333333333333333 * (hi / 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 + (0.3333333333333333d0 * (hi / lo))) ** 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(1.0 + N[(0.3333333333333333 * N[(hi / 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* 15.6%

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

   5. fma-define 15.6%

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

5. Simplified 15.6%

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

   1. add-cube-cbrt 15.6%

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

   2. pow3 15.6%

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

7. Applied egg-rr 15.6%

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

8. Taylor expanded in hi around 0 19.3%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{1 - \frac{x}{lo}} + 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.1111111111111111} \cdot \left(hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right)\right)\right)}}^{3} \]

9. Taylor expanded in x around 0 19.3%

   \[\leadsto \color{blue}{{\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3}} \]
10. 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 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. Taylor expanded in x around 0 18.9%

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

8. Final simplification 18.9%

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

Alternative 6: 18.9% accurate, 0.6× speedup

Math

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

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(hi * Float64(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[(hi * N[(N[(1.0 + N[(hi / 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. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

9. Simplified 18.9%

   \[\leadsto \color{blue}{1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]
10. 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 2024163 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))