xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 24.4s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[Abs[N[(1.0 + N[(hi / lo), $MachinePrecision]), $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. Step-by-step derivation

   1. add-cube-cbrt 18.9%

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

   2. fma-define 18.9%

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

   3. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

   1. add-sqr-sqrt 10.5%

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

   2. sqrt-unprod 19.5%

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

   3. pow2 19.5%

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

   4. fma-undefine 19.5%

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

   5. unpow2 19.5%

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

   6. add-cube-cbrt 19.5%

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

9. Applied egg-rr 19.5%

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

    1. unpow2 19.5%

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

    2. rem-sqrt-square 19.5%

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

    3. +-commutative 19.5%

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

11. Simplified 19.5%

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

12. Final simplification 19.5%

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

Alternative 2: 19.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 8.7%

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

   7. distribute-lft-out 8.9%

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

5. Simplified 8.9%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 17.8%

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

   3. pow2 17.8%

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

   4. associate-/l* 17.8%

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

7. Applied egg-rr 17.8%

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

   1. unpow2 17.8%

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

   2. rem-sqrt-square 17.8%

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

9. Simplified 17.8%

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

10. Taylor expanded in lo around inf 19.1%

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

11. Final simplification 19.1%

    \[\leadsto \left|\left(x - lo\right) \cdot \frac{\frac{lo}{hi}}{hi}\right| \]
12. Add Preprocessing

Alternative 3: 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 4: 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. Final simplification 18.9%

   \[\leadsto 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
10. Add Preprocessing

Alternative 5: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 + N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision] + -1.0), $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. Step-by-step derivation

   1. expm1-log1p-u 18.8%

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

   2. expm1-undefine 18.8%

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

5. Applied egg-rr 18.8%

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

   1. expm1-define 18.8%

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

7. Simplified 18.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi}\right)\right)} \]
8. Step-by-step derivation

   1. expm1-undefine 18.8%

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

   2. log1p-undefine 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Final simplification 18.8%

    \[\leadsto \left(1 + \frac{x - lo}{hi}\right) + -1 \]
11. 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. Final simplification 18.8%

   \[\leadsto \frac{x - lo}{hi} \]
5. 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. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))