xlohi (overflows)

Percentage Accurate: 3.1% → 21.8%

Time: 9.8s

Alternatives: 8

Speedup: 7.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 21.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (expm1 (- (fma (pow (/ lo hi) 2.0) -0.5 (/ x hi)) (/ lo hi))))

C

double code(double lo, double hi, double x) {
	return expm1((fma(pow((lo / hi), 2.0), -0.5, (x / hi)) - (lo / hi)));
}

Julia

function code(lo, hi, x)
	return expm1(Float64(fma((Float64(lo / hi) ^ 2.0), -0.5, Float64(x / hi)) - Float64(lo / hi)))
end

Wolfram

code[lo_, hi_, x_] := N[(Exp[N[(N[(N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] * -0.5 + N[(x / hi), $MachinePrecision]), $MachinePrecision] - N[(lo / hi), $MachinePrecision]), $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. expm1-log1p-u 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\frac{x}{hi} + -0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}\right) - \frac{lo}{hi}}\right) \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   2. *-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\left(\color{blue}{\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} \cdot -0.5} + \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   3. fma-def 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   6. times-frac 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   7. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   8. associate-*r/ 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   9. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   10. associate-*r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   11. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   12. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{hi}{x - lo}}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   13. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   14. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   15. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{hi}{x - lo}}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   16. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   17. pow-sqr 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{\left(2 \cdot -1\right)}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   18. metadata-eval 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{\color{blue}{-2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

7. Simplified 21.7%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-2}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}}\right) \]

8. Taylor expanded in x around 0 0.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{lo}^{2}}{{hi}^{2}}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{lo \cdot lo}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   2. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   3. times-frac 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   4. unpow2 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{lo}{hi}\right)}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

10. Simplified 21.7%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{lo}{hi}\right)}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

11. Final simplification 21.7%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

Alternative 2: 21.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{hi} \cdot \left(x - lo\right)\\ \mathsf{expm1}\left(\mathsf{fma}\left(t_0 \cdot t_0, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 hi) (- x lo))))
   (expm1 (- (fma (* t_0 t_0) -0.5 (/ x hi)) (/ lo hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (1.0 / hi) * (x - lo);
	return expm1((fma((t_0 * t_0), -0.5, (x / hi)) - (lo / hi)));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(1.0 / hi) * Float64(x - lo))
	return expm1(Float64(fma(Float64(t_0 * t_0), -0.5, Float64(x / hi)) - Float64(lo / hi)))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(1.0 / hi), $MachinePrecision] * N[(x - lo), $MachinePrecision]), $MachinePrecision]}, N[(Exp[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -0.5 + N[(x / hi), $MachinePrecision]), $MachinePrecision] - N[(lo / hi), $MachinePrecision]), $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. expm1-log1p-u 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\frac{x}{hi} + -0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}\right) - \frac{lo}{hi}}\right) \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   2. *-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\left(\color{blue}{\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} \cdot -0.5} + \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   3. fma-def 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   6. times-frac 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   7. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   8. associate-*r/ 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   9. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   10. associate-*r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   11. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   12. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{hi}{x - lo}}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   13. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   14. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   15. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{hi}{x - lo}}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   16. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   17. pow-sqr 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{\left(2 \cdot -1\right)}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   18. metadata-eval 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{\color{blue}{-2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

7. Simplified 21.7%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-2}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}}\right) \]
8. Step-by-step derivation

   1. metadata-eval 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{\color{blue}{\left(-1 + -1\right)}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   2. pow-prod-up 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1} \cdot {\left(\frac{hi}{x - lo}\right)}^{-1}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   3. inv-pow 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{hi}{x - lo}}} \cdot {\left(\frac{hi}{x - lo}\right)}^{-1}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   4. inv-pow 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{1}{\frac{hi}{x - lo}} \cdot \color{blue}{\frac{1}{\frac{hi}{x - lo}}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   5. associate-/r/ 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)} \cdot \frac{1}{\frac{hi}{x - lo}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   6. associate-/r/ 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\frac{1}{hi} \cdot \left(x - lo\right)\right) \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

9. Applied egg-rr 21.7%

   \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right) \cdot \left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

10. Final simplification 21.7%

    \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\frac{1}{hi} \cdot \left(x - lo\right)\right) \cdot \left(\frac{1}{hi} \cdot \left(x - lo\right)\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

Alternative 3: 21.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathsf{expm1}\left(-0.5 \cdot {t_0}^{2} + t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi))) (expm1 (+ (* -0.5 (pow t_0 2.0)) t_0))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return expm1(((-0.5 * pow(t_0, 2.0)) + t_0));
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return Math.expm1(((-0.5 * Math.pow(t_0, 2.0)) + t_0));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return math.expm1(((-0.5 * math.pow(t_0, 2.0)) + t_0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return expm1(Float64(Float64(-0.5 * (t_0 ^ 2.0)) + t_0))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(Exp[N[(N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. expm1-log1p-u 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\frac{x}{hi} + -0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}\right) - \frac{lo}{hi}}\right) \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   2. *-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\left(\color{blue}{\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} \cdot -0.5} + \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   3. fma-def 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right)} - \frac{lo}{hi}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   6. times-frac 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   7. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   8. associate-*r/ 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)} \cdot \frac{x - lo}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   9. *-rgt-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{\color{blue}{\left(x - lo\right) \cdot 1}}{hi}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   10. associate-*r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \color{blue}{\left(\left(x - lo\right) \cdot \frac{1}{hi}\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   11. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   12. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{hi}{x - lo}}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   13. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}} \cdot \left(\left(x - lo\right) \cdot \frac{1}{hi}\right), -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   14. *-commutative 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   15. associate-/r/ 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{hi}{x - lo}}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   16. unpow-1 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-1} \cdot \color{blue}{{\left(\frac{hi}{x - lo}\right)}^{-1}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   17. pow-sqr 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{hi}{x - lo}\right)}^{\left(2 \cdot -1\right)}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

   18. metadata-eval 21.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{\color{blue}{-2}}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}\right) \]

7. Simplified 21.7%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{hi}{x - lo}\right)}^{-2}, -0.5, \frac{x}{hi}\right) - \frac{lo}{hi}}\right) \]

8. Taylor expanded in hi around -inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \frac{-1 \cdot x - -1 \cdot lo}{hi} + -0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}}\right) \]
9. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + -1 \cdot \frac{-1 \cdot x - -1 \cdot lo}{hi}}\right) \]

   2. mul-1-neg 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \color{blue}{\left(-\frac{-1 \cdot x - -1 \cdot lo}{hi}\right)}\right) \]

   3. unsub-neg 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} - \frac{-1 \cdot x - -1 \cdot lo}{hi}}\right) \]

   4. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   6. times-frac 21.7%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \color{blue}{\left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right)} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   7. *-lft-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\frac{\color{blue}{1 \cdot \left(x - lo\right)}}{hi} \cdot \frac{x - lo}{hi}\right) - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   8. associate-*l/ 21.7%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)} \cdot \frac{x - lo}{hi}\right) - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   9. *-lft-identity 21.7%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\left(\frac{1}{hi} \cdot \left(x - lo\right)\right) \cdot \frac{\color{blue}{1 \cdot \left(x - lo\right)}}{hi}\right) - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   10. associate-*l/ 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\left(\frac{1}{hi} \cdot \left(x - lo\right)\right) \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}\right) - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   11. unpow1 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\color{blue}{{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}^{1}} \cdot \left(\frac{1}{hi} \cdot \left(x - lo\right)\right)\right) - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   12. pow-plus 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \color{blue}{{\left(\frac{1}{hi} \cdot \left(x - lo\right)\right)}^{\left(1 + 1\right)}} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   13. associate-*l/ 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\color{blue}{\left(\frac{1 \cdot \left(x - lo\right)}{hi}\right)}}^{\left(1 + 1\right)} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   14. *-lft-identity 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{\color{blue}{x - lo}}{hi}\right)}^{\left(1 + 1\right)} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

   15. metadata-eval 21.7%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{\color{blue}{2}} - \frac{-1 \cdot x - -1 \cdot lo}{hi}\right) \]

10. Simplified 21.7%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo - x}{hi}}\right) \]

11. Final simplification 21.7%

    \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2} + \frac{x - lo}{hi}\right) \]

Alternative 4: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{x - lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (/ (- x lo) hi)))

C

double code(double lo, double hi, double x) {
	return log1p(((x - lo) / hi));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(((x - lo) / hi));
}

Python

def code(lo, hi, x):
	return math.log1p(((x - lo) / hi))

Julia

function code(lo, hi, x)
	return log1p(Float64(Float64(x - lo) / hi))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. log1p-expm1-u 18.8%

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

   2. div-inv 18.8%

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

   3. pow2 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot \frac{1}{\color{blue}{{hi}^{2}}}\right)\right)\right) \]

   4. pow-flip 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot \color{blue}{{hi}^{\left(-2\right)}}\right)\right)\right) \]

   5. metadata-eval 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot {hi}^{\color{blue}{-2}}\right)\right)\right) \]

6. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot {hi}^{-2}\right)\right)\right)} \]

7. Taylor expanded in hi around -inf 20.6%

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

8. Final simplification 20.6%

   \[\leadsto \mathsf{log1p}\left(\frac{x - lo}{hi}\right) \]

Alternative 5: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{-lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (/ (- lo) hi)))

C

double code(double lo, double hi, double x) {
	return log1p((-lo / hi));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((-lo / hi));
}

Python

def code(lo, hi, x):
	return math.log1p((-lo / hi))

Julia

function code(lo, hi, x)
	return log1p(Float64(Float64(-lo) / hi))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[((-lo) / hi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. log1p-expm1-u 18.8%

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

   2. div-inv 18.8%

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

   3. pow2 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot \frac{1}{\color{blue}{{hi}^{2}}}\right)\right)\right) \]

   4. pow-flip 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot \color{blue}{{hi}^{\left(-2\right)}}\right)\right)\right) \]

   5. metadata-eval 18.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot {hi}^{\color{blue}{-2}}\right)\right)\right) \]

6. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - x \cdot {hi}^{-2}\right)\right)\right)} \]

7. Taylor expanded in lo around 0 20.6%

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

   1. associate--l+ 20.6%

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

   2. sub-neg 20.6%

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

   3. unpow2 20.6%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{x}{\color{blue}{hi \cdot hi}} + \left(-\frac{1}{hi}\right)\right) \cdot \left(lo \cdot e^{\frac{x}{hi}}\right) + \left(e^{\frac{x}{hi}} - 1\right)\right) \]

   4. distribute-neg-frac 20.6%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{x}{hi \cdot hi} + \color{blue}{\frac{-1}{hi}}\right) \cdot \left(lo \cdot e^{\frac{x}{hi}}\right) + \left(e^{\frac{x}{hi}} - 1\right)\right) \]

   5. metadata-eval 20.6%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{x}{hi \cdot hi} + \frac{\color{blue}{-1}}{hi}\right) \cdot \left(lo \cdot e^{\frac{x}{hi}}\right) + \left(e^{\frac{x}{hi}} - 1\right)\right) \]

   6. expm1-def 20.6%

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

9. Simplified 20.6%

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

10. Taylor expanded in x around 0 20.6%

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

    1. log1p-def 20.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{lo}{hi}\right)} \]

    2. neg-mul-1 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{-\frac{lo}{hi}}\right) \]

    3. distribute-neg-frac 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{-lo}{hi}}\right) \]

12. Simplified 20.6%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{-lo}{hi}\right)} \]

13. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\frac{-lo}{hi}\right) \]

Alternative 6: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / 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. Final simplification 18.8%

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

Alternative 7: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 8: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))