xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 13.0s

Alternatives: 7

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: 19.4% 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 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. associate--l+ 3.1%

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

   3. associate-/l* 15.3%

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

5. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   2. expm1-undefine 15.3%

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

   3. unsub-neg 15.3%

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

   4. fma-neg 18.9%

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

7. Applied egg-rr 18.9%

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

9. Simplified 18.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\right)\right)} \]

10. Taylor expanded in x around inf 18.8%

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

12. Simplified 19.0%

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

13. Taylor expanded in hi around inf 0.0%

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
14. 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 19.7%

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

    4. unpow2 19.7%

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

15. Simplified 19.7%

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

Alternative 2: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(x * N[(N[(N[(1.0 + N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / 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. associate--l+ 3.1%

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

   3. associate-/l* 15.3%

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

5. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   2. expm1-undefine 15.3%

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

   3. unsub-neg 15.3%

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

   4. fma-neg 18.9%

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

7. Applied egg-rr 18.9%

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

9. Simplified 18.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\right)\right)} \]

10. Taylor expanded in x around inf 18.8%

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

12. Simplified 19.0%

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

13. Taylor expanded in lo around inf 19.0%

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

14. Final simplification 19.0%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $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. associate--l+ 3.1%

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

   3. associate-/l* 15.3%

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

5. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   2. expm1-undefine 15.3%

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

   3. unsub-neg 15.3%

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

   4. fma-neg 18.9%

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

7. Applied egg-rr 18.9%

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

9. Simplified 18.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\right)\right)} \]

10. Taylor expanded in x around 0 18.9%

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

    1. sub-neg 18.9%

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

    2. mul-1-neg 18.9%

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

    3. remove-double-neg 18.9%

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

    4. associate-/l* 18.9%

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

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

Alternative 4: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Add Preprocessing

Alternative 5: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

   3. sub-neg 18.7%

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

   4. +-commutative 18.7%

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

   5. distribute-neg-in 18.7%

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

   6. remove-double-neg 18.7%

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

   7. sub-neg 18.7%

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

   8. div-sub 18.7%

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

   9. *-inverses 18.7%

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

5. Simplified 18.7%

   \[\leadsto \color{blue}{1 - \frac{x}{lo}} \]
6. Add Preprocessing

Alternative 6: 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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

8. Simplified 18.7%

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

9. Final simplification 18.7%

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

Alternative 7: 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 18.6%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))