xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 9.7s

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: 18.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, t_0, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {t_0}^{2}, t_0\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)))
   (+
    1.0
    (fma
     (/ hi lo)
     t_0
     (log1p
      (fma
       0.16666666666666666
       (pow (/ hi lo) 3.0)
       (fma 0.5 (pow t_0 2.0) t_0)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return 1.0 + fma((hi / lo), t_0, log1p(fma(0.16666666666666666, pow((hi / lo), 3.0), fma(0.5, pow(t_0, 2.0), t_0))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	return Float64(1.0 + fma(Float64(hi / lo), t_0, log1p(fma(0.16666666666666666, (Float64(hi / lo) ^ 3.0), fma(0.5, (t_0 ^ 2.0), t_0)))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * t$95$0 + N[Log[1 + N[(0.16666666666666666 * N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\left(0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right) - \frac{x}{lo}}\right)\right) \]
8. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(\left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)\right)\right) \]

   4. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} - \frac{x}{lo}\right)}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \color{blue}{\frac{hi - x}{lo}}\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}}, \frac{hi - x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi - x\right) \cdot \left(hi - x\right)}}{{lo}^{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi - x\right) \cdot \left(hi - x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi - x}{lo} \cdot \frac{hi - x}{lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   10. unpow1 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{1}} \cdot \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   11. pow-plus 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{\left(1 + 1\right)}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   12. metadata-eval 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{\color{blue}{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)}\right)\right) \]

10. Taylor expanded in hi around inf 0.0%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{\frac{{hi}^{3}}{{lo}^{3}}}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. cube-div 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

13. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

Alternative 2: 18.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{hi}{lo}\right)}^{2}\\ 1 + \left(t_0 + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, t_0, \frac{hi}{lo}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ hi lo) 2.0)))
   (+
    1.0
    (+
     t_0
     (log1p
      (fma
       0.16666666666666666
       (pow (/ hi lo) 3.0)
       (fma 0.5 t_0 (/ hi lo))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = pow((hi / lo), 2.0);
	return 1.0 + (t_0 + log1p(fma(0.16666666666666666, pow((hi / lo), 3.0), fma(0.5, t_0, (hi / lo)))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(hi / lo) ^ 2.0
	return Float64(1.0 + Float64(t_0 + log1p(fma(0.16666666666666666, (Float64(hi / lo) ^ 3.0), fma(0.5, t_0, Float64(hi / lo))))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]}, N[(1.0 + N[(t$95$0 + N[Log[1 + N[(0.16666666666666666 * N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * t$95$0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\left(0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right) - \frac{x}{lo}}\right)\right) \]
8. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(\left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)\right)\right) \]

   4. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} - \frac{x}{lo}\right)}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \color{blue}{\frac{hi - x}{lo}}\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}}, \frac{hi - x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi - x\right) \cdot \left(hi - x\right)}}{{lo}^{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi - x\right) \cdot \left(hi - x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi - x}{lo} \cdot \frac{hi - x}{lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   10. unpow1 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{1}} \cdot \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   11. pow-plus 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{\left(1 + 1\right)}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   12. metadata-eval 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{\color{blue}{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)}\right)\right) \]

10. Taylor expanded in x around 0 0.0%

    \[\leadsto 1 + \color{blue}{\left(\log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right) + \frac{{hi}^{2}}{{lo}^{2}}\right)} \]
11. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto 1 + \color{blue}{\left(\frac{{hi}^{2}}{{lo}^{2}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right)} \]

    2. unpow2 0.0%

       \[\leadsto 1 + \left(\frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    3. unpow2 0.0%

       \[\leadsto 1 + \left(\frac{hi \cdot hi}{\color{blue}{lo \cdot lo}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    4. times-frac 0.0%

       \[\leadsto 1 + \left(\color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    5. unpow2 0.0%

       \[\leadsto 1 + \left(\color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    6. log1p-def 0.0%

       \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \color{blue}{\mathsf{log1p}\left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)}\right) \]

    7. fma-def 0.0%

       \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{hi}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)}\right)\right) \]

    8. cube-div 0.0%

       \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right) \]

    9. fma-def 0.0%

       \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, \frac{hi}{lo}\right)}\right)\right)\right) \]

    10. unpow2 0.0%

        \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    11. unpow2 0.0%

        \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    12. times-frac 19.1%

        \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    13. unpow2 19.1%

        \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}}, \frac{hi}{lo}\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \color{blue}{\left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo}\right)}^{2}, \frac{hi}{lo}\right)\right)\right)\right)} \]

13. Final simplification 19.1%

    \[\leadsto 1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo}\right)}^{2}, \frac{hi}{lo}\right)\right)\right)\right) \]

Alternative 3: 18.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ 1 + \left(\frac{hi}{lo} \cdot t_0 + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {t_0}^{3}, \mathsf{fma}\left(0.5, {t_0}^{2}, t_0\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)))
   (+
    1.0
    (+
     (* (/ hi lo) t_0)
     (log1p
      (fma 0.16666666666666666 (pow t_0 3.0) (fma 0.5 (pow t_0 2.0) t_0)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return 1.0 + (((hi / lo) * t_0) + log1p(fma(0.16666666666666666, pow(t_0, 3.0), fma(0.5, pow(t_0, 2.0), t_0))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	return Float64(1.0 + Float64(Float64(Float64(hi / lo) * t_0) + log1p(fma(0.16666666666666666, (t_0 ^ 3.0), fma(0.5, (t_0 ^ 2.0), t_0)))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, N[(1.0 + N[(N[(N[(hi / lo), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Log[1 + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision] + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\left(0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right) - \frac{x}{lo}}\right)\right) \]
8. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(\left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)\right)\right) \]

   4. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} - \frac{x}{lo}\right)}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \color{blue}{\frac{hi - x}{lo}}\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}}, \frac{hi - x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi - x\right) \cdot \left(hi - x\right)}}{{lo}^{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi - x\right) \cdot \left(hi - x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi - x}{lo} \cdot \frac{hi - x}{lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   10. unpow1 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{1}} \cdot \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   11. pow-plus 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{\left(1 + 1\right)}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   12. metadata-eval 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{\color{blue}{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. fma-udef 19.1%

       \[\leadsto 1 + \color{blue}{\left(\frac{hi}{lo} \cdot \frac{hi - x}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right)} \]

11. Applied egg-rr 19.1%

    \[\leadsto 1 + \color{blue}{\left(\frac{hi}{lo} \cdot \frac{hi - x}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right)} \]

12. Final simplification 19.1%

    \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi - x}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

Alternative 4: 18.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, t_0, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {t_0}^{3}, t_0 + 0.5 \cdot {t_0}^{2}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)))
   (+
    1.0
    (fma
     (/ hi lo)
     t_0
     (log1p
      (fma
       0.16666666666666666
       (pow t_0 3.0)
       (+ t_0 (* 0.5 (pow t_0 2.0)))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return 1.0 + fma((hi / lo), t_0, log1p(fma(0.16666666666666666, pow(t_0, 3.0), (t_0 + (0.5 * pow(t_0, 2.0))))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	return Float64(1.0 + fma(Float64(hi / lo), t_0, log1p(fma(0.16666666666666666, (t_0 ^ 3.0), Float64(t_0 + Float64(0.5 * (t_0 ^ 2.0)))))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * t$95$0 + N[Log[1 + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision] + N[(t$95$0 + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\left(0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right) - \frac{x}{lo}}\right)\right) \]
8. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{0.16666666666666666 \cdot \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}} + \left(\left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi - x\right)}^{3}}{{lo}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)}\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{3}}, \left(0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right) - \frac{x}{lo}\right)\right)\right) \]

   4. associate--l+ 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} - \frac{x}{lo}\right)}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}} + \color{blue}{\frac{hi - x}{lo}}\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi - x\right)}^{2}}{{lo}^{2}}, \frac{hi - x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi - x\right) \cdot \left(hi - x\right)}}{{lo}^{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi - x\right) \cdot \left(hi - x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi - x}{lo} \cdot \frac{hi - x}{lo}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   10. unpow1 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{1}} \cdot \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   11. pow-plus 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi - x}{lo}\right)}^{\left(1 + 1\right)}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

   12. metadata-eval 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{\color{blue}{2}}, \frac{hi - x}{lo}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. fma-udef 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot {\left(\frac{hi - x}{lo}\right)}^{2} + \frac{hi - x}{lo}}\right)\right)\right) \]

11. Applied egg-rr 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \color{blue}{0.5 \cdot {\left(\frac{hi - x}{lo}\right)}^{2} + \frac{hi - x}{lo}}\right)\right)\right) \]

12. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi - x}{lo}\right)}^{3}, \frac{hi - x}{lo} + 0.5 \cdot {\left(\frac{hi - x}{lo}\right)}^{2}\right)\right)\right) \]

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in hi around inf 18.9%

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

6. Taylor expanded in lo around 0 0.0%

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

   1. unpow2 0.0%

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

   2. times-frac 18.9%

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

   3. *-commutative 18.9%

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

   4. distribute-rgt1-in 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

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. 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 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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023292 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))