sintan (problem 3.4.5)

Percentage Accurate: 1.6% → 99.9%

Time: 19.4s

Alternatives: 4

Speedup: 207.0×

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, 10.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = ((eps * eps) * (0.225d0 + (eps * (eps * ((-0.009642857142857142d0) + ((eps * eps) * 0.00024107142857142857d0)))))) + (-0.5d0)
end function

Java

public static double code(double eps) {
	return ((eps * eps) * (0.225 + (eps * (eps * (-0.009642857142857142 + ((eps * eps) * 0.00024107142857142857)))))) + -0.5;
}

Python

def code(eps):
	return ((eps * eps) * (0.225 + (eps * (eps * (-0.009642857142857142 + ((eps * eps) * 0.00024107142857142857)))))) + -0.5

Julia

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

MATLAB

function tmp = code(eps)
	tmp = ((eps * eps) * (0.225 + (eps * (eps * (-0.009642857142857142 + ((eps * eps) * 0.00024107142857142857)))))) + -0.5;
end

Wolfram

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

Derivation

1. Initial program 2.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

5. Simplified 99.9%

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

Alternative 2: 99.9% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (-0.5d0) + ((eps * eps) * (0.225d0 + (eps * (eps * (-0.009642857142857142d0)))))
end function

Java

public static double code(double eps) {
	return -0.5 + ((eps * eps) * (0.225 + (eps * (eps * -0.009642857142857142))));
}

Python

def code(eps):
	return -0.5 + ((eps * eps) * (0.225 + (eps * (eps * -0.009642857142857142))))

Julia

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

MATLAB

function tmp = code(eps)
	tmp = -0.5 + ((eps * eps) * (0.225 + (eps * (eps * -0.009642857142857142))));
end

Wolfram

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

Derivation

1. Initial program 2.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \left(\frac{-27}{2800} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \left({\varepsilon}^{2} \cdot \frac{-27}{2800}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-27}{2800}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-27}{2800}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{-27}{2800}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-27}{2800}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   12. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{9}{40}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-27}{2800}\right)\right)\right)\right), \frac{-1}{2}\right) \]

5. Simplified 99.7%

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

6. Final simplification 99.7%

   \[\leadsto -0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.225 + \varepsilon \cdot \left(\varepsilon \cdot -0.009642857142857142\right)\right) \]
7. Add Preprocessing

Alternative 3: 99.7% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{9}{40} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{9}{40}, \left({\varepsilon}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{9}{40}, \left(\varepsilon \cdot \varepsilon\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{9}{40}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   6. metadata-eval 99.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{9}{40}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \frac{-1}{2}\right) \]

5. Simplified 99.1%

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

6. Final simplification 99.1%

   \[\leadsto -0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.225 \]
7. Add Preprocessing

Alternative 4: 99.1% accurate, 207.0× 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 2.6%

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

3. Taylor expanded in eps around 0

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

5. Simplified 98.3%

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

Developer Target 1: 99.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \left(\left(-0.5 + \frac{9 \cdot \left(\varepsilon \cdot \varepsilon\right)}{40}\right) + \frac{-27 \cdot t\_0}{2800}\right) + \frac{27 \cdot \left(\left(t\_0 \cdot \varepsilon\right) \cdot \varepsilon\right)}{112000} \end{array} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (let* ((t_0 (* (* (* eps eps) eps) eps)))
   (+
    (+ (+ -0.5 (/ (* 9.0 (* eps eps)) 40.0)) (/ (* -27.0 t_0) 2800.0))
    (/ (* 27.0 (* (* t_0 eps) eps)) 112000.0))))

C

double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = ((eps * eps) * eps) * eps
    code = (((-0.5d0) + ((9.0d0 * (eps * eps)) / 40.0d0)) + (((-27.0d0) * t_0) / 2800.0d0)) + ((27.0d0 * ((t_0 * eps) * eps)) / 112000.0d0)
end function

Java

public static double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

Python

def code(eps):
	t_0 = ((eps * eps) * eps) * eps
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0)

Julia

function code(eps)
	t_0 = Float64(Float64(Float64(eps * eps) * eps) * eps)
	return Float64(Float64(Float64(-0.5 + Float64(Float64(9.0 * Float64(eps * eps)) / 40.0)) + Float64(Float64(-27.0 * t_0) / 2800.0)) + Float64(Float64(27.0 * Float64(Float64(t_0 * eps) * eps)) / 112000.0))
end

MATLAB

function tmp = code(eps)
	t_0 = ((eps * eps) * eps) * eps;
	tmp = ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
end

Wolfram

code[eps_] := Block[{t$95$0 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, N[(N[(N[(-0.5 + N[(N[(9.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / 40.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-27.0 * t$95$0), $MachinePrecision] / 2800.0), $MachinePrecision]), $MachinePrecision] + N[(N[(27.0 * N[(N[(t$95$0 * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] / 112000.0), $MachinePrecision]), $MachinePrecision]]

Developer Target 2: 99.7% accurate, 29.6× 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 2024191 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))

  :alt
  (! :herbie-platform default (+ -1/2 (/ (* 9 (* eps eps)) 40) (/ (* -27 (* eps eps eps eps)) 2800) (/ (* 27 (* eps eps eps eps eps eps)) 112000)))

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

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