xlohi (overflows)

Percentage Accurate: 3.1% → 22.2%

Time: 25.6s

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: 22.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + {t\_0}^{3}}{\left(1 - t\_0\right) + {\left(\frac{\mathsf{fma}\left(hi, -1, x\right)}{lo}\right)}^{2}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (fma hi (+ -1.0 (/ (- x hi) lo)) x) lo)))
   (expm1
    (log1p
     (/
      (+ 1.0 (pow t_0 3.0))
      (+ (- 1.0 t_0) (pow (/ (fma hi -1.0 x) lo) 2.0)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = fma(hi, (-1.0 + ((x - hi) / lo)), x) / lo;
	return expm1(log1p(((1.0 + pow(t_0, 3.0)) / ((1.0 - t_0) + pow((fma(hi, -1.0, x) / lo), 2.0)))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(fma(hi, Float64(-1.0 + Float64(Float64(x - hi) / lo)), x) / lo)
	return expm1(log1p(Float64(Float64(1.0 + (t_0 ^ 3.0)) / Float64(Float64(1.0 - t_0) + (Float64(fma(hi, -1.0, x) / lo) ^ 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

   1. +-commutative 14.8%

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

   2. flip3-+ 13.9%

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

7. Applied egg-rr 13.5%

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

   1. +-commutative 13.5%

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

   2. associate-+r- 13.5%

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

   3. associate-/r/ 13.5%

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

   4. associate-+r- 13.5%

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

   5. associate-/r/ 13.5%

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

   6. *-rgt-identity 13.5%

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

9. Simplified 13.5%

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

    1. expm1-log1p-u 13.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + {\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{3}}{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2} + \left(1 - \frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}\right)\right)} \]

    2. expm1-undefine 13.5%

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

11. Applied egg-rr 13.5%

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

13. Simplified 19.1%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right)}^{3}}{\left(1 - \frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right) + {\left(\frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right)}^{2}}\right)\right)} \]

14. Taylor expanded in lo around inf 22.2%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right)}^{3}}{\left(1 - \frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right) + {\left(\frac{\mathsf{fma}\left(hi, \color{blue}{-1}, x\right)}{lo}\right)}^{2}}\right)\right) \]
15. Add Preprocessing

Alternative 2: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ -1 + e^{\mathsf{log1p}\left(1 + \frac{x + \mathsf{fma}\left(\frac{hi}{lo}, x - hi, -hi\right)}{lo}\right)} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ -1.0 (exp (log1p (+ 1.0 (/ (+ x (fma (/ hi lo) (- x hi) (- hi))) lo))))))

C

double code(double lo, double hi, double x) {
	return -1.0 + exp(log1p((1.0 + ((x + fma((hi / lo), (x - hi), -hi)) / lo))));
}

Julia

function code(lo, hi, x)
	return Float64(-1.0 + exp(log1p(Float64(1.0 + Float64(Float64(x + fma(Float64(hi / lo), Float64(x - hi), Float64(-hi))) / lo)))))
end

Wolfram

code[lo_, hi_, x_] := N[(-1.0 + N[Exp[N[Log[1 + N[(1.0 + N[(N[(x + N[(N[(hi / lo), $MachinePrecision] * N[(x - hi), $MachinePrecision] + (-hi)), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

   1. add-cube-cbrt 14.8%

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

7. Applied egg-rr 13.7%

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

   1. unpow2 13.7%

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

   2. add-cube-cbrt 13.7%

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

   3. expm1-log1p-u 13.5%

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

   4. expm1-undefine 13.5%

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

   5. associate-/r/ 13.5%

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

   6. fmm-def 19.1%

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

9. Applied egg-rr 19.1%

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

10. Final simplification 19.1%

    \[\leadsto -1 + e^{\mathsf{log1p}\left(1 + \frac{x + \mathsf{fma}\left(\frac{hi}{lo}, x - hi, -hi\right)}{lo}\right)} \]
11. Add Preprocessing

Alternative 3: 19.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

   1. add-cube-cbrt 14.8%

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

7. Applied egg-rr 13.7%

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

   1. unpow2 13.7%

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

   2. add-cube-cbrt 13.7%

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

   3. +-commutative 13.7%

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

   4. associate-/r/ 13.7%

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

   5. fmm-def 19.1%

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

9. Applied egg-rr 19.1%

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

10. Final simplification 19.1%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

6. Taylor expanded in hi around 0 18.9%

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

   1. sub-neg 18.9%

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

   2. +-commutative 18.9%

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

   3. neg-mul-1 18.9%

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

   4. sub-neg 18.9%

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

   5. div-sub 18.9%

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

   6. metadata-eval 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

6. Taylor expanded in hi around 0 18.9%

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

   1. sub-neg 18.9%

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

   2. +-commutative 18.9%

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

   3. neg-mul-1 18.9%

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

   4. sub-neg 18.9%

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

   5. div-sub 18.9%

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

   6. metadata-eval 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in x around 0 18.9%

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

    1. associate-/l* 18.9%

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

11. Simplified 18.9%

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

Alternative 6: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in hi around inf 18.8%

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

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(lo / Float64(-hi))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
10. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))