sintan (problem 3.4.5)

Percentage Accurate: 1.6% → 99.9%

Time: 16.7s

Alternatives: 5

Speedup: 218.0×

Specification

\[-0.4 \leq \varepsilon \land \varepsilon \leq 0.4\]

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 1.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\\ t_1 := \mathsf{fma}\left(t\_0, \varepsilon \cdot \varepsilon, 0.5\right)\\ t_2 := {t\_1}^{2}\\ \frac{t\_2 \cdot \left({t\_0}^{4} \cdot {\varepsilon}^{8}\right) - 0.0625 \cdot t\_2}{\left(\mathsf{fma}\left({\varepsilon}^{4}, {t\_0}^{2}, 0.25\right) \cdot t\_1\right) \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* eps eps) 0.00024107142857142857 -0.009642857142857142)
          (* eps eps)
          0.225))
        (t_1 (fma t_0 (* eps eps) 0.5))
        (t_2 (pow t_1 2.0)))
   (/
    (- (* t_2 (* (pow t_0 4.0) (pow eps 8.0))) (* 0.0625 t_2))
    (* (* (fma (pow eps 4.0) (pow t_0 2.0) 0.25) t_1) t_2))))

C

double code(double eps) {
	double t_0 = fma(fma((eps * eps), 0.00024107142857142857, -0.009642857142857142), (eps * eps), 0.225);
	double t_1 = fma(t_0, (eps * eps), 0.5);
	double t_2 = pow(t_1, 2.0);
	return ((t_2 * (pow(t_0, 4.0) * pow(eps, 8.0))) - (0.0625 * t_2)) / ((fma(pow(eps, 4.0), pow(t_0, 2.0), 0.25) * t_1) * t_2);
}

Julia

function code(eps)
	t_0 = fma(fma(Float64(eps * eps), 0.00024107142857142857, -0.009642857142857142), Float64(eps * eps), 0.225)
	t_1 = fma(t_0, Float64(eps * eps), 0.5)
	t_2 = t_1 ^ 2.0
	return Float64(Float64(Float64(t_2 * Float64((t_0 ^ 4.0) * (eps ^ 8.0))) - Float64(0.0625 * t_2)) / Float64(Float64(fma((eps ^ 4.0), (t_0 ^ 2.0), 0.25) * t_1) * t_2))
end

Wolfram

code[eps_] := Block[{t$95$0 = N[(N[(N[(eps * eps), $MachinePrecision] * 0.00024107142857142857 + -0.009642857142857142), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.225), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(N[(t$95$2 * N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[Power[eps, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) \cdot {\varepsilon}^{2}} + \frac{9}{40}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right)}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-27}{2800}}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{27}{112000}, {\varepsilon}^{2}, \frac{-27}{2800}\right)}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-27}{2800}\right), {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-27}{2800}\right), {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   16. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, \color{blue}{-0.5}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \frac{\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\right)}^{2} \cdot {\varepsilon}^{4}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)}} \]

9. Applied rewrites 100.0%

   \[\leadsto \frac{\left({\varepsilon}^{8} \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\right)}^{4}\right) \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2} - 0.0625 \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2}}{\color{blue}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \mathsf{fma}\left({\varepsilon}^{4}, {\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\right)}^{2}, 0.25\right)\right)}} \]

10. Final simplification 100.0%

    \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2} \cdot \left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\right)}^{4} \cdot {\varepsilon}^{8}\right) - 0.0625 \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2}}{\left(\mathsf{fma}\left({\varepsilon}^{4}, {\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right)\right)}^{2}, 0.25\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right) \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, 0.5\right)\right)}^{2}} \]
11. Add Preprocessing

Alternative 2: 99.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma
  (fma
   (fma 0.00024107142857142857 (* eps eps) -0.009642857142857142)
   (* eps eps)
   0.225)
  (* eps eps)
  -0.5))

C

double code(double eps) {
	return fma(fma(fma(0.00024107142857142857, (eps * eps), -0.009642857142857142), (eps * eps), 0.225), (eps * eps), -0.5);
}

Julia

function code(eps)
	return fma(fma(fma(0.00024107142857142857, Float64(eps * eps), -0.009642857142857142), Float64(eps * eps), 0.225), Float64(eps * eps), -0.5)
end

Wolfram

code[eps_] := N[(N[(N[(0.00024107142857142857 * N[(eps * eps), $MachinePrecision] + -0.009642857142857142), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.225), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) \cdot {\varepsilon}^{2}} + \frac{9}{40}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right)}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-27}{2800}}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{27}{112000}, {\varepsilon}^{2}, \frac{-27}{2800}\right)}, {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-27}{2800}\right), {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-27}{2800}\right), {\varepsilon}^{2}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{27}{112000}, \varepsilon \cdot \varepsilon, \frac{-27}{2800}\right), \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   16. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, \color{blue}{-0.5}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00024107142857142857, \varepsilon \cdot \varepsilon, -0.009642857142857142\right), \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right)} \]
6. Add Preprocessing

Alternative 3: 99.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma (fma -0.009642857142857142 (* eps eps) 0.225) (* eps eps) -0.5))

C

double code(double eps) {
	return fma(fma(-0.009642857142857142, (eps * eps), 0.225), (eps * eps), -0.5);
}

Julia

function code(eps)
	return fma(fma(-0.009642857142857142, Float64(eps * eps), 0.225), Float64(eps * eps), -0.5)
end

Wolfram

code[eps_] := N[(N[(-0.009642857142857142 * N[(eps * eps), $MachinePrecision] + 0.225), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-27}{2800} \cdot {\varepsilon}^{2} + \frac{9}{40}}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right)}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}\right), {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right)} \]
6. Add Preprocessing

Alternative 4: 99.7% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, -0.5\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (fma (* eps eps) 0.225 -0.5))

C

double code(double eps) {
	return fma((eps * eps), 0.225, -0.5);
}

Julia

function code(eps)
	return fma(Float64(eps * eps), 0.225, -0.5)
end

Wolfram

code[eps_] := N[(N[(eps * eps), $MachinePrecision] * 0.225 + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. metadata-eval 99.7

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, -0.5\right)} \]
6. Add Preprocessing

Alternative 5: 99.1% accurate, 218.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 -0.5)

C

double code(double eps) {
	return -0.5;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = -0.5d0
end function

Java

public static double code(double eps) {
	return -0.5;
}

Python

def code(eps):
	return -0.5

Julia

function code(eps)
	return -0.5
end

MATLAB

function tmp = code(eps)
	tmp = -0.5;
end

Wolfram

code[eps_] := -0.5

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{-1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{-0.5} \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \left(0.225 \cdot \varepsilon\right) \cdot \varepsilon - 0.5 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (- (* (* 0.225 eps) eps) 0.5))

C

double code(double eps) {
	return ((0.225 * eps) * eps) - 0.5;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = ((0.225d0 * eps) * eps) - 0.5d0
end function

Java

public static double code(double eps) {
	return ((0.225 * eps) * eps) - 0.5;
}

Python

def code(eps):
	return ((0.225 * eps) * eps) - 0.5

Julia

function code(eps)
	return Float64(Float64(Float64(0.225 * eps) * eps) - 0.5)
end

MATLAB

function tmp = code(eps)
	tmp = ((0.225 * eps) * eps) - 0.5;
end

Wolfram

code[eps_] := N[(N[(N[(0.225 * eps), $MachinePrecision] * eps), $MachinePrecision] - 0.5), $MachinePrecision]

Reproduce

herbie shell --seed 2024296 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))

  :alt
  (! :herbie-platform default (- (* 9/40 eps eps) 1/2))

  (/ (- eps (sin eps)) (- eps (tan eps))))