Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 16.8s

Precision: binary64

Cost: 26240

Math

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]

↓

\[y + \mathsf{fma}\left(z, \log \left(\frac{e}{t}\right), \mathsf{fma}\left(a + -0.5, b, x\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (+ y (fma z (log (/ E t)) (fma (+ a -0.5) b x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return y + fma(z, log((((double) M_E) / t)), fma((a + -0.5), b, x));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
end

↓

function code(x, y, z, t, a, b)
	return Float64(y + fma(z, log(Float64(exp(1) / t)), fma(Float64(a + -0.5), b, x)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := N[(y + N[(z * N[Log[N[(E / t), $MachinePrecision]], $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.1
Target	0.4
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

Derivation

1. Initial program 0.1

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]

2. Simplified 0.1

   \[\leadsto \color{blue}{y + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, x\right)\right)} \]
   Proof
   (+.f64 y (fma.f64 z (-.f64 1 (log.f64 t)) (fma.f64 (+.f64 a -1/2) b x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (fma.f64 z (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (log.f64 t)))) (fma.f64 (+.f64 a -1/2) b x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (fma.f64 z (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (log.f64 t)) 1)) (fma.f64 (+.f64 a -1/2) b x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (fma.f64 z (+.f64 (neg.f64 (log.f64 t)) 1) (fma.f64 (+.f64 a (Rewrite<= metadata-eval (neg.f64 1/2))) b x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (fma.f64 z (+.f64 (neg.f64 (log.f64 t)) 1) (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 a 1/2)) b x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (fma.f64 z (+.f64 (neg.f64 (log.f64 t)) 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 a 1/2) b) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (+.f64 (neg.f64 (log.f64 t)) 1)) (+.f64 (*.f64 (-.f64 a 1/2) b) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (neg.f64 (log.f64 t)) 1) z)) (+.f64 (*.f64 (-.f64 a 1/2) b) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 z (*.f64 (neg.f64 (log.f64 t)) z))) (+.f64 (*.f64 (-.f64 a 1/2) b) x))): 14 points increase in error, 1 points decrease in error
   (+.f64 y (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 z (*.f64 (log.f64 t) z))) (+.f64 (*.f64 (-.f64 a 1/2) b) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (-.f64 z (Rewrite<= *-commutative_binary64 (*.f64 z (log.f64 t)))) (+.f64 (*.f64 (-.f64 a 1/2) b) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (Rewrite=> associate-+l-_binary64 (-.f64 z (-.f64 (*.f64 z (log.f64 t)) (+.f64 (*.f64 (-.f64 a 1/2) b) x))))): 1 points increase in error, 0 points decrease in error
   (+.f64 y (-.f64 z (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b)) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 z (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b))) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (-.f64 z (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 z (log.f64 t)) (*.f64 (neg.f64 (-.f64 a 1/2)) b)))) x)): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (Rewrite=> associate--r+_binary64 (-.f64 (-.f64 z (*.f64 z (log.f64 t))) (*.f64 (neg.f64 (-.f64 a 1/2)) b))) x)): 0 points increase in error, 1 points decrease in error
   (+.f64 y (Rewrite=> associate-+l-_binary64 (-.f64 (-.f64 z (*.f64 z (log.f64 t))) (-.f64 (*.f64 (neg.f64 (-.f64 a 1/2)) b) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (-.f64 z (*.f64 z (log.f64 t))) (*.f64 (neg.f64 (-.f64 a 1/2)) b)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (Rewrite<= associate--r+_binary64 (-.f64 z (+.f64 (*.f64 z (log.f64 t)) (*.f64 (neg.f64 (-.f64 a 1/2)) b)))) x)): 1 points increase in error, 0 points decrease in error
   (+.f64 y (+.f64 (-.f64 z (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b)))) x)): 0 points increase in error, 0 points decrease in error
   (+.f64 y (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 z (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 y x) (-.f64 z (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 x y)) (-.f64 z (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 x y) z) (-.f64 (*.f64 z (log.f64 t)) (*.f64 (-.f64 a 1/2) b)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a 1/2) b))): 0 points increase in error, 1 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto y + \mathsf{fma}\left(z, \color{blue}{\log \left(\frac{e}{t}\right)}, \mathsf{fma}\left(a + -0.5, b, x\right)\right) \]

4. Final simplification 0.1

   \[\leadsto y + \mathsf{fma}\left(z, \log \left(\frac{e}{t}\right), \mathsf{fma}\left(a + -0.5, b, x\right)\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	19904

\[y + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, x\right)\right) \]

Alternative 2
Error	0.1
Cost	7360

\[\left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) + b \cdot \left(a + -0.5\right) \]

Alternative 3
Error	11.0
Cost	7248

\[\begin{array}{l} t_1 := \left(z + \left(y + x\right)\right) + b \cdot \left(a + -0.5\right)\\ t_2 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -7.957361945696984 \cdot 10^{+229}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.309809269377338 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.368873225820934 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.734564894876411 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	11.0
Cost	7248

\[\begin{array}{l} t_1 := \left(z + \left(y + x\right)\right) + b \cdot \left(a + -0.5\right)\\ t_2 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -7.957361945696984 \cdot 10^{+229}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.309809269377338 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.368873225820934 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.734564894876411 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	9.2
Cost	7112

\[\begin{array}{l} t_1 := x + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -7.368873225820934 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.734564894876411 \cdot 10^{+164}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.9
Cost	7112

\[\begin{array}{l} t_1 := x + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -7.368873225820934 \cdot 10^{+104}:\\ \;\;\;\;y + t_1\\ \mathbf{elif}\;z \leq 7.734564894876411 \cdot 10^{+164}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	37.2
Cost	1240

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ t_2 := z + t_1\\ \mathbf{if}\;x \leq -1.519209433235064 \cdot 10^{+169}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.9456712574625998 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.102154439782116 \cdot 10^{+90}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.3922257745236152 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.937989269784567 \cdot 10^{-30}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;x \leq 8.260166438081532 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 8
Error	36.1
Cost	1112

\[\begin{array}{l} t_1 := y + -0.5 \cdot b\\ \mathbf{if}\;x \leq -1.519209433235064 \cdot 10^{+169}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.9456712574625998 \cdot 10^{+141}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -3.0979979443249815 \cdot 10^{-34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -4.372040017977281 \cdot 10^{-78}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -9.191135023346093 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.175954560451962 \cdot 10^{-266}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	36.0
Cost	1112

\[\begin{array}{l} t_1 := y + -0.5 \cdot b\\ \mathbf{if}\;x \leq -1.519209433235064 \cdot 10^{+169}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.9456712574625998 \cdot 10^{+141}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -3.0979979443249815 \cdot 10^{-34}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;x \leq -4.372040017977281 \cdot 10^{-78}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -9.191135023346093 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.175954560451962 \cdot 10^{-266}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.3
Cost	1096

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+177}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	36.5
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.519209433235064 \cdot 10^{+169}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.9456712574625998 \cdot 10^{+141}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -3.0979979443249815 \cdot 10^{-34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.4994422896073765 \cdot 10^{-79}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -5.239469094375334 \cdot 10^{-115}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq 1.1180995783023715 \cdot 10^{-246}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 12
Error	46.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.519209433235064 \cdot 10^{+169}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.9456712574625998 \cdot 10^{+141}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -3.0979979443249815 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1180995783023715 \cdot 10^{-246}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 13
Error	15.1
Cost	704

\[\left(z + \left(y + x\right)\right) + b \cdot \left(a + -0.5\right) \]

Alternative 14
Error	26.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.8520332402366603 \cdot 10^{+133}:\\ \;\;\;\;x + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 15
Error	25.9
Cost	580

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;y \leq 4.937679217572032 \cdot 10^{+32}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 16
Error	43.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 8.016134672785661 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 17
Error	61.5
Cost	64

\[z \]

Alternative 18
Error	48.0
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))