xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 19.1s

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: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

   1. add-cbrt-cube 18.9%

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

   2. pow3 18.9%

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

   3. +-commutative 18.9%

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

   4. fma-define 18.9%

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

7. Applied egg-rr 18.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{hi}{lo} + 1, \frac{hi - x}{lo}, 1\right)\right)}^{3}}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 10.1%

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

   2. sqrt-unprod 19.6%

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

   3. pow2 19.6%

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

9. Applied egg-rr 19.6%

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

    1. unpow2 19.6%

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

    2. rem-sqrt-square 19.6%

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

    3. +-commutative 19.6%

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

11. Simplified 19.6%

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

12. Taylor expanded in lo around 0 19.6%

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

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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 + (abs((1.0d0 + (hi / lo))) / (lo / (hi - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[Abs[N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(lo / N[(hi - 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 0.0%

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

5. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. un-div-inv 18.9%

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

7. Applied egg-rr 18.9%

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

   1. add-sqr-sqrt 10.1%

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

   2. sqrt-unprod 19.6%

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

   3. pow2 19.6%

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

9. Applied egg-rr 19.6%

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

    1. unpow2 19.6%

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

    2. rem-sqrt-square 19.6%

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

    3. +-commutative 19.6%

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

11. Simplified 19.6%

    \[\leadsto 1 + \frac{\color{blue}{\left|1 + \frac{hi}{lo}\right|}}{\frac{lo}{hi - x}} \]
12. Add Preprocessing

Alternative 3: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 - \left|hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in 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* 15.3%

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

   5. fma-define 15.3%

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

5. Simplified 15.3%

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

   1. add-sqr-sqrt 10.1%

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

   2. sqrt-unprod 14.8%

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

   3. pow2 14.8%

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

7. Applied egg-rr 14.8%

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

   1. unpow2 14.8%

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

   2. rem-sqrt-square 14.8%

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

9. Simplified 19.6%

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

10. Taylor expanded in x around 0 19.6%

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

    1. mul-1-neg 19.6%

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

    2. associate-*r/ 19.6%

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

    3. distribute-rgt-neg-in 19.6%

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

    4. distribute-frac-neg 19.6%

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

    5. mul-1-neg 19.6%

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

    6. distribute-lft-in 19.6%

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

    7. metadata-eval 19.6%

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

    8. mul-1-neg 19.6%

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

    9. unsub-neg 19.6%

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

12. Simplified 19.6%

    \[\leadsto 1 - \left|\color{blue}{hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}}\right| \]
13. Add Preprocessing

Alternative 4: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{hi}{lo}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ hi lo) 2.0))

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 2.0);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi / lo) ** 2.0d0
end function

Java

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

Python

def code(lo, hi, x):
	return math.pow((hi / lo), 2.0)

Julia

function code(lo, hi, x)
	return Float64(hi / lo) ^ 2.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi / lo) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

   1. add-cbrt-cube 18.9%

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

   2. pow3 18.9%

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

   3. +-commutative 18.9%

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

   4. fma-define 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around inf 18.9%

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

   1. +-commutative 18.9%

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

   2. distribute-lft-in 18.9%

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

   3. rgt-mult-inverse 18.9%

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

   4. fma-undefine 18.9%

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

10. Simplified 18.9%

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

11. Taylor expanded in hi around inf 0.0%

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
12. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

    2. unpow2 0.0%

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

    3. times-frac 19.4%

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

    4. unpow2 19.4%

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

13. Simplified 19.4%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
14. Add Preprocessing

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $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 0.0%

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

   \[\leadsto 1 + \left(1 + \frac{hi}{lo}\right) \cdot \frac{hi - x}{lo} \]
7. 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 lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

   \[\leadsto 1 + \color{blue}{hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]
9. 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 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

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 2024088 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))