xlohi (overflows)

Percentage Accurate: 3.1% → 19.2%

Time: 23.8s

Alternatives: 7

Speedup: 7.0×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 19.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{hi}\right)}^{2}\\ \sqrt{{\left(1 + \frac{x + \left(\mathsf{fma}\left(hi \cdot {\left(\sqrt[3]{lo}\right)}^{-2}, \frac{x - hi}{\sqrt[3]{lo}}, \sqrt[3]{hi} \cdot \left(-t\_0\right)\right) + \mathsf{fma}\left(-\sqrt[3]{hi}, t\_0, \sqrt[3]{hi} \cdot e^{2 \cdot \log \left(\sqrt[3]{hi}\right)}\right)\right)}{lo}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (cbrt hi) 2.0)))
   (sqrt
    (pow
     (+
      1.0
      (/
       (+
        x
        (+
         (fma
          (* hi (pow (cbrt lo) -2.0))
          (/ (- x hi) (cbrt lo))
          (* (cbrt hi) (- t_0)))
         (fma (- (cbrt hi)) t_0 (* (cbrt hi) (exp (* 2.0 (log (cbrt hi))))))))
       lo))
     2.0))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow(cbrt(hi), 2.0);
	return sqrt(pow((1.0 + ((x + (fma((hi * pow(cbrt(lo), -2.0)), ((x - hi) / cbrt(lo)), (cbrt(hi) * -t_0)) + fma(-cbrt(hi), t_0, (cbrt(hi) * exp((2.0 * log(cbrt(hi)))))))) / lo)), 2.0));
}

Julia

function code(lo, hi, x)
	t_0 = cbrt(hi) ^ 2.0
	return sqrt((Float64(1.0 + Float64(Float64(x + Float64(fma(Float64(hi * (cbrt(lo) ^ -2.0)), Float64(Float64(x - hi) / cbrt(lo)), Float64(cbrt(hi) * Float64(-t_0))) + fma(Float64(-cbrt(hi)), t_0, Float64(cbrt(hi) * exp(Float64(2.0 * log(cbrt(hi)))))))) / lo)) ^ 2.0))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[Power[hi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[(N[(N[(hi * N[Power[N[Power[lo, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(x - hi), $MachinePrecision] / N[Power[lo, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[hi, 1/3], $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[hi, 1/3], $MachinePrecision]) * t$95$0 + N[(N[Power[hi, 1/3], $MachinePrecision] * N[Exp[N[(2.0 * N[Log[N[Power[hi, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.0%

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

5. Simplified 15.0%

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

   1. add-sqr-sqrt 15.0%

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

   2. sqrt-unprod 15.0%

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

   3. pow2 15.0%

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

7. Applied egg-rr 19.2%

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

   1. *-un-lft-identity 19.2%

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

   2. add-cube-cbrt 19.2%

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

   3. unpow2 19.2%

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

   4. frac-times 19.2%

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

   5. fmm-def 14.6%

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

   6. associate-*r* 14.6%

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

   7. add-cube-cbrt 14.6%

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

   8. prod-diff 19.2%

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

9. Applied egg-rr 19.2%

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

    1. add-exp-log 19.2%

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

    2. log-pow 19.2%

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

11. Applied egg-rr 19.2%

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

12. Final simplification 19.2%

    \[\leadsto \sqrt{{\left(1 + \frac{x + \left(\mathsf{fma}\left(hi \cdot {\left(\sqrt[3]{lo}\right)}^{-2}, \frac{x - hi}{\sqrt[3]{lo}}, \sqrt[3]{hi} \cdot \left(-{\left(\sqrt[3]{hi}\right)}^{2}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{hi}, {\left(\sqrt[3]{hi}\right)}^{2}, \sqrt[3]{hi} \cdot e^{2 \cdot \log \left(\sqrt[3]{hi}\right)}\right)\right)}{lo}\right)}^{2}} \]
13. Add Preprocessing

Alternative 2: 19.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{hi}\right)}^{2}\\ \sqrt{{\left(1 + \frac{x + \left(\mathsf{fma}\left(\frac{x - hi}{lo}, hi, \sqrt[3]{hi} \cdot \left(-t\_0\right)\right) + \mathsf{fma}\left(-\sqrt[3]{hi}, t\_0, \sqrt[3]{hi} \cdot t\_0\right)\right)}{lo}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (cbrt hi) 2.0)))
   (sqrt
    (pow
     (+
      1.0
      (/
       (+
        x
        (+
         (fma (/ (- x hi) lo) hi (* (cbrt hi) (- t_0)))
         (fma (- (cbrt hi)) t_0 (* (cbrt hi) t_0))))
       lo))
     2.0))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow(cbrt(hi), 2.0);
	return sqrt(pow((1.0 + ((x + (fma(((x - hi) / lo), hi, (cbrt(hi) * -t_0)) + fma(-cbrt(hi), t_0, (cbrt(hi) * t_0)))) / lo)), 2.0));
}

Julia

function code(lo, hi, x)
	t_0 = cbrt(hi) ^ 2.0
	return sqrt((Float64(1.0 + Float64(Float64(x + Float64(fma(Float64(Float64(x - hi) / lo), hi, Float64(cbrt(hi) * Float64(-t_0))) + fma(Float64(-cbrt(hi)), t_0, Float64(cbrt(hi) * t_0)))) / lo)) ^ 2.0))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[Power[hi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[(N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * hi + N[(N[Power[hi, 1/3], $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[hi, 1/3], $MachinePrecision]) * t$95$0 + N[(N[Power[hi, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.0%

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

5. Simplified 15.0%

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

   1. add-sqr-sqrt 15.0%

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

   2. sqrt-unprod 15.0%

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

   3. pow2 15.0%

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

7. Applied egg-rr 19.2%

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

   1. fmm-undef 14.6%

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

   2. *-commutative 14.6%

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

   3. add-cube-cbrt 14.6%

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

   4. prod-diff 19.2%

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

   5. pow2 19.2%

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

   6. pow2 19.2%

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

   7. pow2 19.2%

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

9. Applied egg-rr 19.2%

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

10. Final simplification 19.2%

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

Alternative 3: 19.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(1 + \frac{x + {\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}\right)}^{3}}{lo}\right)}^{2}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (sqrt
  (pow
   (+ 1.0 (/ (+ x (pow (cbrt (fma hi (/ (- x hi) lo) (- hi))) 3.0)) lo))
   2.0)))

C

double code(double lo, double hi, double x) {
	return sqrt(pow((1.0 + ((x + pow(cbrt(fma(hi, ((x - hi) / lo), -hi)), 3.0)) / lo)), 2.0));
}

Julia

function code(lo, hi, x)
	return sqrt((Float64(1.0 + Float64(Float64(x + (cbrt(fma(hi, Float64(Float64(x - hi) / lo), Float64(-hi))) ^ 3.0)) / lo)) ^ 2.0))
end

Wolfram

code[lo_, hi_, x_] := N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[Power[N[Power[N[(hi * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + (-hi)), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.0%

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

5. Simplified 15.0%

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

   1. add-sqr-sqrt 15.0%

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

   2. sqrt-unprod 15.0%

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

   3. pow2 15.0%

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

7. Applied egg-rr 19.2%

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

   1. *-un-lft-identity 19.2%

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

   2. add-cube-cbrt 19.2%

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

   3. unpow2 19.2%

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

   4. frac-times 19.2%

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

   5. fmm-def 14.6%

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

   6. add-cube-cbrt 14.6%

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

   7. pow3 14.6%

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

9. Applied egg-rr 19.2%

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

Alternative 4: 19.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.0%

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

5. Simplified 15.0%

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

   1. add-sqr-sqrt 15.0%

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

   2. sqrt-unprod 15.0%

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

   3. pow2 15.0%

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

7. Applied egg-rr 19.2%

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

   1. pow1/2 19.2%

      \[\leadsto \color{blue}{{\left({\left(1 + \frac{x + \mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}{lo}\right)}^{2}\right)}^{0.5}} \]

9. Applied egg-rr 19.2%

   \[\leadsto \color{blue}{{\left({\left(1 + \frac{x + \mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}{lo}\right)}^{2}\right)}^{0.5}} \]
10. Step-by-step derivation

    1. unpow1/2 19.2%

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

    2. unpow2 19.2%

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

    3. rem-sqrt-square 19.2%

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

11. Simplified 19.2%

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

12. Final simplification 19.2%

    \[\leadsto \left|1 + \frac{\mathsf{fma}\left(hi, -1 - \frac{hi - x}{lo}, x\right)}{lo}\right| \]
13. Add Preprocessing

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.0%

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

5. Simplified 15.0%

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

6. Taylor expanded in hi around 0 18.9%

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

   1. sub-neg 18.9%

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

   2. neg-mul-1 18.9%

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

   3. +-commutative 18.9%

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

   4. sub-neg 18.9%

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

   5. div-sub 18.9%

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

   6. metadata-eval 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 6: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(lo, hi, x)
	return Float64(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. Add Preprocessing

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-*r/ 18.9%

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

   2. neg-mul-1 18.9%

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

8. Simplified 18.9%

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

Alternative 7: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))