VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 99.3%

Time: 4.8s

Alternatives: 9

Speedup: 4.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \ell\\ t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (PI) l))) (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))

Initial Program: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \ell\\ t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (PI) l))) (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (* l_s (if (<= l_m 6.8e+14) (- (* (PI) l_m) (/ (/ (tan t_0) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 6.8e14

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   4. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1}{-F} \cdot \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{-F}} \]

   6. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1 \cdot \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{-F}}{-F}} \]

   8. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}} \]

   if 6.8e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+263} \lor \neg \left(t\_1 \leq -1 \cdot 10^{-210}\right):\\ \;\;\;\;l\_m \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)) (t_1 (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))
   (*
    l_s
    (if (or (<= t_1 -5e+263) (not (<= t_1 -1e-210)))
      (* l_m (PI))
      (* (- l_m) (/ (PI) (* F F)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -5.00000000000000022e263 or -1e-210 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 65.2%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   if -5.00000000000000022e263 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1e-210

   1. Initial program 98.2%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\frac{-1}{F \cdot F} \cdot \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

   6. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]

   8. Applied rewrites 25.0%

      \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -5 \cdot 10^{+263} \lor \neg \left(\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -1 \cdot 10^{-210}\right):\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 90000000000000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m - \frac{-1}{F} \cdot \left(\mathsf{fma}\left(\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{F} \cdot 0.3333333333333333, l\_m \cdot l\_m, \frac{\mathsf{PI}\left(\right)}{F}\right) \cdot \left(-l\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 90000000000000.0)
    (-
     (* (PI) l_m)
     (*
      (/ -1.0 F)
      (*
       (fma
        (* (/ (* (* (PI) (PI)) (PI)) F) 0.3333333333333333)
        (* l_m l_m)
        (/ (PI) F))
       (- l_m))))
    (* l_m (PI)))))

Derivation

1. Split input into 2 regimes

2. if l < 9e13

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   4. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1}{-F} \cdot \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{-F}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{-F} \cdot \color{blue}{\left(\ell \cdot \left(-1 \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)\right) + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)\right)} \]

   7. Applied rewrites 71.7%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{-F} \cdot \color{blue}{\left(\left(-1 \cdot \mathsf{fma}\left(\frac{{\mathsf{PI}\left(\right)}^{3}}{F} \cdot 0.3333333333333333, \ell \cdot \ell, \frac{\mathsf{PI}\left(\right)}{F}\right)\right) \cdot \ell\right)} \]

   9. Applied rewrites 71.7%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{-F} \cdot \left(\left(-1 \cdot \mathsf{fma}\left(\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{F} \cdot 0.3333333333333333, \ell \cdot \ell, \frac{\mathsf{PI}\left(\right)}{F}\right)\right) \cdot \ell\right) \]

   if 9e13 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 90000000000000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{-1}{F} \cdot \left(\mathsf{fma}\left(\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{F} \cdot 0.3333333333333333, \ell \cdot \ell, \frac{\mathsf{PI}\left(\right)}{F}\right) \cdot \left(-\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (* l_s (if (<= l_m 2.4e+14) (- (* (PI) l_m) (/ (/ t_0 F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2.4e14

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   4. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1}{-F} \cdot \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{-F}} \]

   6. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1 \cdot \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{-F}}{-F}} \]

   8. Applied rewrites 87.5%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}}{F}}{F} \]

   11. Applied rewrites 83.5%

       \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}}{F}}{F} \]

   if 2.4e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m - \frac{t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (* l_s (if (<= l_m 2.4e+14) (- (* (PI) l_m) (/ t_0 (* F F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2.4e14

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

   5. Applied rewrites 75.3%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

   7. Applied rewrites 75.6%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1 \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F \cdot F}} \]

   if 2.4e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \mathsf{PI}\left(\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m - l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 2.4e+14)
    (- (* (PI) l_m) (* l_m (/ (PI) (* F F))))
    (* l_m (PI)))))

Derivation

1. Split input into 2 regimes

2. if l < 2.4e14

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]

   5. Applied rewrites 75.8%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}} \]

   if 2.4e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 2.4e+14) (* (- (PI) (/ (PI) (* F F))) l_m) (* l_m (PI)))))

Derivation

1. Split input into 2 regimes

2. if l < 2.4e14

   1. Initial program 78.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot \ell} \]

   if 2.4e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.5% accurate, 22.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(l\_m \cdot \mathsf{PI}\left(\right)\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* l_m (PI))))

Derivation

1. Initial program 74.8%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
2. Add Preprocessing

3. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

5. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
6. Add Preprocessing

Alternative 9: 3.1% accurate, 135.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot 0 \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s 0.0))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Fortran

l\_m =     private
l\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_s, f, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_s
    real(8), intent (in) :: f
    real(8), intent (in) :: l_m
    code = l_s * 0.0d0
end function

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * 0.0

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * 0.0)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * 0.0;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
2. Add Preprocessing

4. Applied rewrites 55.3%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\tan \left(\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)} \]

5. Taylor expanded in l around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)}} \]

7. Applied rewrites 2.5%

   \[\leadsto \color{blue}{\frac{0}{F \cdot F}} \]

8. Taylor expanded in F around 0

   \[\leadsto 0 \]

10. Applied rewrites 3.0%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025042 -o generate:proofs
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))