xlohi (overflows)

Percentage Accurate: 3.1% → 97.9%

Time: 10.8s

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: 97.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) - 1 \cdot 1\right), \color{blue}{\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo} - 1\right)}\right) \]

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo} + hi \cdot \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right)\right)}, 1\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\left(hi \cdot \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right) + -1 \cdot \frac{x}{lo}\right), 1\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\left(hi \cdot \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right) + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right), 1\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\left(hi \cdot \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right) - \frac{x}{lo}\right), 1\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(hi \cdot \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \left(-1 \cdot \frac{x}{{lo}^{2}} + \frac{1}{lo}\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \left(\frac{1}{lo} + -1 \cdot \frac{x}{{lo}^{2}}\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \left(\frac{1}{lo} + \left(\mathsf{neg}\left(\frac{x}{{lo}^{2}}\right)\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   8. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\left(\frac{1}{lo}\right), \left(\frac{x}{{lo}^{2}}\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \left(\frac{x}{{lo}^{2}}\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \left({lo}^{2}\right)\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \left(lo \cdot lo\right)\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \left(\frac{x}{lo}\right)\right), 1\right)\right) \]

   14. /-lowering-/.f64 26.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(lo, \mathsf{\_.f64}\left(hi, x\right)\right)\right)\right)\right), 1\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, lo\right)\right), 1\right)\right) \]

10. Simplified 26.7%

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

11. Taylor expanded in lo around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{-1}, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, lo\right)\right), 1\right)\right) \]
12. Step-by-step derivation

13. Simplified 96.8%

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

14. Final simplification 96.8%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi\right), \color{blue}{lo}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\frac{hi \cdot \left(x - hi\right)}{lo} - hi\right)\right), lo\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{hi \cdot \left(x - hi\right)}{lo} - hi\right)\right), lo\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{hi \cdot \left(x - hi\right)}{lo} + \left(\mathsf{neg}\left(hi\right)\right)\right)\right), lo\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + \left(\mathsf{neg}\left(hi\right)\right)\right)\right), lo\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + -1 \cdot hi\right)\right), lo\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + hi \cdot -1\right)\right), lo\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(\frac{x - hi}{lo} + -1\right)\right)\right), lo\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(\frac{x - hi}{lo} + -1\right)\right)\right), lo\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\left(\frac{x - hi}{lo}\right), -1\right)\right)\right), lo\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(x - hi\right), lo\right), -1\right)\right)\right), lo\right)\right) \]

   15. --lowering--.f64 18.9%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, hi\right), lo\right), -1\right)\right)\right), lo\right)\right) \]

5. Simplified 18.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \left(hi \cdot \left(1 + \frac{hi}{lo}\right)\right)\right)}\right), lo\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(hi \cdot \left(1 + \frac{hi}{lo}\right)\right)\right)\right), lo\right)\right) \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(\mathsf{neg}\left(\left(1 + \frac{hi}{lo}\right)\right)\right)\right)\right), lo\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(\mathsf{neg}\left(\left(\frac{hi}{lo} + 1\right)\right)\right)\right)\right), lo\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(\left(\mathsf{neg}\left(\frac{hi}{lo}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), lo\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(-1 \cdot \frac{hi}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), lo\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(-1 \cdot \frac{hi}{lo} - 1\right)\right)\right), lo\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 \cdot \frac{hi}{lo} - 1\right)\right)\right), lo\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 \cdot \frac{hi}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), lo\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 \cdot \frac{hi}{lo} + -1\right)\right)\right), lo\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 + -1 \cdot \frac{hi}{lo}\right)\right)\right), lo\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 + \left(\mathsf{neg}\left(\frac{hi}{lo}\right)\right)\right)\right)\right), lo\right)\right) \]

   12. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(-1 - \frac{hi}{lo}\right)\right)\right), lo\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(-1, \left(\frac{hi}{lo}\right)\right)\right)\right), lo\right)\right) \]

   14. /-lowering-/.f64 18.9%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), lo\right)\right) \]

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi\right), \color{blue}{lo}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\frac{hi \cdot \left(x - hi\right)}{lo} - hi\right)\right), lo\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{hi \cdot \left(x - hi\right)}{lo} - hi\right)\right), lo\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{hi \cdot \left(x - hi\right)}{lo} + \left(\mathsf{neg}\left(hi\right)\right)\right)\right), lo\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + \left(\mathsf{neg}\left(hi\right)\right)\right)\right), lo\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + -1 \cdot hi\right)\right), lo\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \frac{x - hi}{lo} + hi \cdot -1\right)\right), lo\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(hi \cdot \left(\frac{x - hi}{lo} + -1\right)\right)\right), lo\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \left(\frac{x - hi}{lo} + -1\right)\right)\right), lo\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\left(\frac{x - hi}{lo}\right), -1\right)\right)\right), lo\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(x - hi\right), lo\right), -1\right)\right)\right), lo\right)\right) \]

   15. --lowering--.f64 18.9%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, hi\right), lo\right), -1\right)\right)\right), lo\right)\right) \]

5. Simplified 18.9%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{hi \cdot \left(1 + \frac{hi}{lo}\right)}{lo}\right)}\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(hi \cdot \color{blue}{\frac{1 + \frac{hi}{lo}}{lo}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(hi, \color{blue}{\left(\frac{1 + \frac{hi}{lo}}{lo}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\left(1 + \frac{hi}{lo}\right), \color{blue}{lo}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{hi}{lo}\right)\right), lo\right)\right)\right) \]

   9. /-lowering-/.f64 18.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), lo\right)\right)\right) \]

8. Simplified 18.9%

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

Alternative 4: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.7%

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

6. Taylor expanded in lo around inf

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

   1. associate-/l/ N/A

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

   2. div-sub N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \color{blue}{\left(\frac{\frac{x}{lo} - 1}{hi}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(\frac{x}{lo} - 1\right), \color{blue}{hi}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(\frac{x}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right), hi\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(\frac{x}{lo} + -1\right), hi\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(-1 + \frac{x}{lo}\right), hi\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \left(\frac{x}{lo}\right)\right), hi\right)\right) \]

   9. /-lowering-/.f64 18.8%

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x, lo\right)\right), hi\right)\right) \]

8. Simplified 18.8%

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

Alternative 5: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{hi}\right), \color{blue}{\left(lo \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \left(\color{blue}{lo} \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \color{blue}{\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \color{blue}{\left(-1 \cdot \frac{lo}{{hi}^{2}} - \frac{1}{hi}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \left(-1 \cdot \frac{lo}{{hi}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(-1 \cdot \frac{lo}{{hi}^{2}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)}\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{lo}{{hi}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(0 - \frac{lo}{{hi}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{lo}{{hi}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \left({hi}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{hi}}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \left(hi \cdot hi\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)\right)\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{hi}}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\frac{-1}{hi}\right)\right)\right) \]

   12. /-lowering-/.f64 18.8%

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{hi}\right)\right)\right) \]

8. Simplified 18.8%

   \[\leadsto \color{blue}{lo \cdot \left(\left(0 - \frac{lo}{hi \cdot hi}\right) + \frac{-1}{hi}\right)} \]

9. Taylor expanded in lo around 0

   \[\leadsto \mathsf{*.f64}\left(lo, \color{blue}{\left(\frac{-1}{hi}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f64 18.8%

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(-1, \color{blue}{hi}\right)\right) \]

11. Simplified 18.8%

    \[\leadsto lo \cdot \color{blue}{\frac{-1}{hi}} \]
12. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in lo around inf

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

5. Simplified 18.7%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))