xlohi (overflows)

Percentage Accurate: 3.1% → 19.3%

Time: 8.5s

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, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 9.4%

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

   1. associate--l+ 9.4%

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

   2. distribute-lft-out-- 9.4%

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

   3. div-sub 9.4%

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

   4. mul-1-neg 9.4%

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

   5. unsub-neg 9.4%

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

4. Simplified 9.4%

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

   1. add-sqr-sqrt 8.6%

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

   2. sqrt-unprod 18.0%

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

   3. pow2 18.0%

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

6. Applied egg-rr 18.0%

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

   1. unpow2 18.0%

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

   2. rem-sqrt-square 18.0%

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

   3. div-sub 18.0%

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

   4. associate-+l- 18.0%

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

   5. sub-neg 18.0%

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

   6. associate-+r+ 18.0%

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

   7. +-commutative 18.0%

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

   8. sub-neg 18.0%

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

   9. div-sub 18.0%

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

8. Simplified 18.0%

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

9. Taylor expanded in lo around 0 19.3%

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

10. Final simplification 19.3%

    \[\leadsto \left|\frac{hi - x}{lo}\right| \]

Alternative 2: 19.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 9.4%

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

   1. associate--l+ 9.4%

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

   2. distribute-lft-out-- 9.4%

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

   3. div-sub 9.4%

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

   4. mul-1-neg 9.4%

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

   5. unsub-neg 9.4%

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

4. Simplified 9.4%

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

   1. add-sqr-sqrt 8.6%

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

   2. sqrt-unprod 18.0%

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

   3. pow2 18.0%

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

6. Applied egg-rr 18.0%

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

   1. unpow2 18.0%

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

   2. rem-sqrt-square 18.0%

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

   3. div-sub 18.0%

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

   4. associate-+l- 18.0%

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

   5. sub-neg 18.0%

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

   6. associate-+r+ 18.0%

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

   7. +-commutative 18.0%

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

   8. sub-neg 18.0%

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

   9. div-sub 18.0%

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

8. Simplified 18.0%

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

9. Taylor expanded in lo around 0 19.3%

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

10. Taylor expanded in hi around inf 19.3%

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

11. Final simplification 19.3%

    \[\leadsto \left|\frac{hi}{lo}\right| \]

Alternative 3: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

   2. sub-neg 18.9%

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

   3. add-sqr-sqrt 9.9%

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

   4. sqrt-unprod 13.7%

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

   5. sqr-neg 13.7%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in hi around -inf 0.0%

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

   1. +-commutative 0.0%

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

   2. mul-1-neg 0.0%

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

   3. unsub-neg 0.0%

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

   4. unpow2 0.0%

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

   5. unpow2 0.0%

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

   6. times-frac 18.9%

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

   7. sub-neg 18.9%

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

   8. unpow2 18.9%

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

   9. distribute-neg-frac 18.9%

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

   10. metadata-eval 18.9%

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

9. Simplified 18.9%

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

    1. *-un-lft-identity 18.9%

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

    2. times-frac 18.9%

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

11. Applied egg-rr 18.9%

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

    1. associate-*l/ 18.9%

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

    2. *-lft-identity 18.9%

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

13. Simplified 18.9%

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

14. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ 1 + \left(t_0 + t_0 \cdot \frac{hi}{lo}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo))) (+ 1.0 (+ t_0 (* t_0 (/ hi lo))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return 1.0 + (t_0 + (t_0 * (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
    real(8) :: t_0
    t_0 = (hi - x) / lo
    code = 1.0d0 + (t_0 + (t_0 * (hi / lo)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return 1.0 + (t_0 + (t_0 * (hi / lo)));
}

Python

def code(lo, hi, x):
	t_0 = (hi - x) / lo
	return 1.0 + (t_0 + (t_0 * (hi / lo)))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	return Float64(1.0 + Float64(t_0 + Float64(t_0 * Float64(hi / lo))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi - x) / lo;
	tmp = 1.0 + (t_0 + (t_0 * (hi / lo)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, N[(1.0 + N[(t$95$0 + N[(t$95$0 * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in lo around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. unpow2 0.0%

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

   4. times-frac 18.9%

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

   5. div-sub 18.9%

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

7. Simplified 18.9%

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

8. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. fma-udef 18.9%

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

   2. sub-neg 18.9%

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

   3. add-sqr-sqrt 9.9%

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

   4. sqrt-unprod 14.4%

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

   5. sqr-neg 14.4%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 18.9%

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

   8. sub-neg 18.9%

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

   9. add-sqr-sqrt 9.9%

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

   10. sqrt-unprod 13.7%

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

   11. sqr-neg 13.7%

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

   12. sqrt-unprod 9.0%

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

   13. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

   1. distribute-lft1-in 18.9%

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

   2. +-commutative 18.9%

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

   3. +-commutative 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 6: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

   2. sub-neg 18.9%

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

   3. add-sqr-sqrt 9.9%

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

   4. sqrt-unprod 13.7%

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

   5. sqr-neg 13.7%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in x around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. unpow2 0.0%

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

   4. times-frac 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

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

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. Taylor expanded in hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))