xlohi (overflows)

Percentage Accurate: 3.1% → 21.3%

Time: 7.5s

Alternatives: 5

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: 21.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] * 0.5 + N[(lo / hi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(x / hi), $MachinePrecision] - N[Log[N[(1.0 + N[(N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(x / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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%

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

   2. log1p-udef 18.8%

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

   3. inv-pow 18.8%

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

   4. div-inv 18.8%

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

   5. pow2 18.8%

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

   6. pow-flip 18.8%

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

   7. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate-+r+ 0.0%

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

   3. mul-1-neg 0.0%

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

   4. unsub-neg 0.0%

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

   5. *-commutative 0.0%

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

   6. fma-def 0.0%

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

   7. unpow2 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 11.5%

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

   10. unpow2 11.5%

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

   11. *-commutative 11.5%

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

   12. unpow2 11.5%

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

   13. times-frac 21.4%

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

9. Simplified 21.4%

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

    1. add-cube-cbrt 21.4%

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

    2. pow2 21.4%

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

11. Applied egg-rr 21.4%

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

12. Final simplification 21.4%

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

Alternative 2: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - \log \left(1 + \left(\left(\frac{lo}{hi} + {\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5\right) - \frac{x}{hi} \cdot \frac{lo}{hi}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (/ x hi)
  (log
   (+
    1.0
    (- (+ (/ lo hi) (* (pow (/ lo hi) 2.0) 0.5)) (* (/ x hi) (/ lo hi)))))))

C

double code(double lo, double hi, double x) {
	return (x / hi) - log((1.0 + (((lo / hi) + (pow((lo / hi), 2.0) * 0.5)) - ((x / hi) * (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 / hi) - log((1.0d0 + (((lo / hi) + (((lo / hi) ** 2.0d0) * 0.5d0)) - ((x / hi) * (lo / hi)))))
end function

Java

public static double code(double lo, double hi, double x) {
	return (x / hi) - Math.log((1.0 + (((lo / hi) + (Math.pow((lo / hi), 2.0) * 0.5)) - ((x / hi) * (lo / hi)))));
}

Python

def code(lo, hi, x):
	return (x / hi) - math.log((1.0 + (((lo / hi) + (math.pow((lo / hi), 2.0) * 0.5)) - ((x / hi) * (lo / hi)))))

Julia

function code(lo, hi, x)
	return Float64(Float64(x / hi) - log(Float64(1.0 + Float64(Float64(Float64(lo / hi) + Float64((Float64(lo / hi) ^ 2.0) * 0.5)) - Float64(Float64(x / hi) * Float64(lo / hi))))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x / hi) - log((1.0 + (((lo / hi) + (((lo / hi) ^ 2.0) * 0.5)) - ((x / hi) * (lo / hi)))));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[Log[N[(1.0 + N[(N[(N[(lo / hi), $MachinePrecision] + N[(N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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%

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

   2. log1p-udef 18.8%

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

   3. inv-pow 18.8%

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

   4. div-inv 18.8%

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

   5. pow2 18.8%

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

   6. pow-flip 18.8%

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

   7. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate-+r+ 0.0%

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

   3. mul-1-neg 0.0%

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

   4. unsub-neg 0.0%

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

   5. *-commutative 0.0%

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

   6. fma-def 0.0%

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

   7. unpow2 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 11.5%

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

   10. unpow2 11.5%

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

   11. *-commutative 11.5%

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

   12. unpow2 11.5%

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

   13. times-frac 21.4%

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

9. Simplified 21.4%

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

    1. fma-udef 21.4%

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

11. Applied egg-rr 21.4%

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

12. Final simplification 21.4%

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

Alternative 3: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(lo, hi, x):
	return math.log((1.0 + ((x - lo) / hi)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.4%

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

   6. div-sub 9.4%

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

4. Simplified 9.4%

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

   1. add-log-exp 9.4%

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

   2. fma-def 9.4%

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

6. Applied egg-rr 9.4%

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

7. Taylor expanded in hi around inf 20.6%

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

   1. sub-neg 20.6%

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

   2. +-commutative 20.6%

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

   3. neg-mul-1 20.6%

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

   4. associate-+l+ 20.6%

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

   5. neg-mul-1 20.6%

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

   6. sub-neg 20.6%

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

   7. div-sub 20.6%

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

9. Simplified 20.6%

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

10. Final simplification 20.6%

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

Alternative 4: 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 5: 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 2023213 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))