VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 99.1%

Time: 5.2s

Alternatives: 8

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: 76.5% 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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - {F}^{-1} \cdot \frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 9.5e+14)
      (- t_0 (* (pow F -1.0) (/ (tan (* l_m (PI))) F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 9.5e14

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 88.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 9.5e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - \frac{\frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (* l_s (if (<= l_m 9.5e+14) (- t_0 (/ (/ (tan (* l_m (PI))) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 9.5e14

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 88.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 9.5e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

Alternative 3: 98.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{F}, \frac{\mathsf{PI}\left(\right)}{F} \cdot l\_m, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 8e+14)
    (fma (/ -1.0 F) (* (/ (PI) F) l_m) (* l_m (PI)))
    (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 8e14

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 88.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - {F}^{-1} \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.7%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - {F}^{-1} \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{F} \cdot \ell\right)} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 83.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

   if 8e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

Alternative 4: 98.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - \frac{\frac{\mathsf{PI}\left(\right)}{F} \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (* l_s (if (<= l_m 8e+14) (- t_0 (/ (* (/ (PI) F) l_m) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 8e14

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 88.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.7%

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

   if 8e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\mathsf{PI}\left(\right)}{F} \cdot \ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\frac{\mathsf{PI}\left(\right)}{F}}{F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 8e+14) (* (- (PI) (/ (/ (PI) F) F)) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 8e14

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 88.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 77.7%

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

   if 8e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\frac{\mathsf{PI}\left(\right)}{F}}{F}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 8e+14) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 8e14

   1. Initial program 82.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.7%

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

   if 8e14 < l

   1. Initial program 54.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

Alternative 7: 73.1% accurate, 22.5× speedup

Math

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

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 (* (PI) l_m)))

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

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

5. Applied rewrites 76.7%

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

Alternative 8: 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
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. lower-tan.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-fma.f64 58.0

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

4. Applied rewrites 58.0%

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

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 3.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 3.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))