xlohi (overflows)

Percentage Accurate: 3.1% → 98.1%

Time: 14.1s

Alternatives: 6

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: 98.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 - N[(N[(x / lo), $MachinePrecision] * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - N[(x / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / 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 18.8%

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

   10. distribute-lft-out-- 18.8%

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

   11. associate-*r/ 18.8%

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

   12. fma-neg 18.8%

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

4. Simplified 18.8%

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

   1. flip-+ 18.8%

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

   2. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

   1. fma-def 18.8%

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

   2. distribute-lft1-in 18.8%

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

   3. +-commutative 18.8%

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

   4. fma-def 18.8%

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

   5. distribute-lft1-in 18.8%

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

   6. +-commutative 18.8%

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

   7. fma-def 18.8%

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

   8. distribute-lft1-in 18.8%

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

   9. +-commutative 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in hi around 0 26.5%

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

    1. +-commutative 26.5%

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

    2. fma-def 26.5%

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

    3. +-commutative 26.5%

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

    4. mul-1-neg 26.5%

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

    5. unsub-neg 26.5%

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

    6. unpow2 26.5%

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

    7. mul-1-neg 26.5%

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

11. Simplified 26.5%

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

12. Taylor expanded in hi around 0 74.3%

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

    1. unpow2 74.3%

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

    2. unpow2 74.3%

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

    3. times-frac 97.5%

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

14. Simplified 97.5%

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

15. Final simplification 97.5%

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

Alternative 2: 27.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ t_1 := \left(1 + \frac{hi}{lo}\right) \cdot t_0\\ \frac{1 - t_1 \cdot t_1}{1 - t_0} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)) (t_1 (* (+ 1.0 (/ hi lo)) t_0)))
   (/ (- 1.0 (* t_1 t_1)) (- 1.0 t_0))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double t_1 = (1.0 + (hi / lo)) * t_0;
	return (1.0 - (t_1 * t_1)) / (1.0 - t_0);
}

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
    real(8) :: t_1
    t_0 = (hi - x) / lo
    t_1 = (1.0d0 + (hi / lo)) * t_0
    code = (1.0d0 - (t_1 * t_1)) / (1.0d0 - t_0)
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double t_1 = (1.0 + (hi / lo)) * t_0;
	return (1.0 - (t_1 * t_1)) / (1.0 - t_0);
}

Python

def code(lo, hi, x):
	t_0 = (hi - x) / lo
	t_1 = (1.0 + (hi / lo)) * t_0
	return (1.0 - (t_1 * t_1)) / (1.0 - t_0)

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	t_1 = Float64(Float64(1.0 + Float64(hi / lo)) * t_0)
	return Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 - t_0))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi - x) / lo;
	t_1 = (1.0 + (hi / lo)) * t_0;
	tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_0);
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $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 18.8%

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

   10. distribute-lft-out-- 18.8%

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

   11. associate-*r/ 18.8%

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

   12. fma-neg 18.8%

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

4. Simplified 18.8%

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

   1. flip-+ 18.8%

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

   2. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

   1. fma-def 18.8%

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

   2. distribute-lft1-in 18.8%

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

   3. +-commutative 18.8%

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

   4. fma-def 18.8%

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

   5. distribute-lft1-in 18.8%

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

   6. +-commutative 18.8%

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

   7. fma-def 18.8%

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

   8. distribute-lft1-in 18.8%

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

   9. +-commutative 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in lo around inf 26.5%

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

10. Final simplification 26.5%

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

Alternative 3: 27.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(hi / 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 18.8%

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

   10. distribute-lft-out-- 18.8%

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

   11. associate-*r/ 18.8%

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

   12. fma-neg 18.8%

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

4. Simplified 18.8%

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

   1. flip-+ 18.8%

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

   2. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

   1. fma-def 18.8%

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

   2. distribute-lft1-in 18.8%

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

   3. +-commutative 18.8%

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

   4. fma-def 18.8%

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

   5. distribute-lft1-in 18.8%

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

   6. +-commutative 18.8%

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

   7. fma-def 18.8%

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

   8. distribute-lft1-in 18.8%

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

   9. +-commutative 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in hi around 0 26.5%

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

    1. +-commutative 26.5%

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

    2. fma-def 26.5%

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

    3. +-commutative 26.5%

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

    4. mul-1-neg 26.5%

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

    5. unsub-neg 26.5%

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

    6. unpow2 26.5%

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

    7. mul-1-neg 26.5%

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

11. Simplified 26.5%

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

12. Taylor expanded in x around 0 26.5%

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

13. Final simplification 26.5%

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

Alternative 4: 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. 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 18.8%

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

   10. distribute-lft-out-- 18.8%

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

   11. associate-*r/ 18.8%

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

   12. fma-neg 18.8%

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

4. Simplified 18.8%

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

   1. flip-+ 18.8%

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

   2. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

   1. fma-def 18.8%

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

   2. distribute-lft1-in 18.8%

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

   3. +-commutative 18.8%

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

   4. fma-def 18.8%

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

   5. distribute-lft1-in 18.8%

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

   6. +-commutative 18.8%

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

   7. fma-def 18.8%

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

   8. distribute-lft1-in 18.8%

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

   9. +-commutative 18.8%

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

8. Simplified 18.8%

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

   1. metadata-eval 18.8%

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

   2. flip-+ 18.8%

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

   3. *-commutative 18.8%

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

   4. +-commutative 18.8%

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

10. Applied egg-rr 18.8%

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

11. Final simplification 18.8%

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

Alternative 5: 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 6: 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 2023278 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))