Average Error: 0.1 → 0.1
Time: 17.7s
Precision: binary64
Cost: 19904
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
↓
\[\mathsf{fma}\left(z, 1 - \log t, y\right) + \mathsf{fma}\left(a + -0.5, b, x\right) \]
(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
 (+ (fma z (- 1.0 (log t)) y) (fma (+ a -0.5) b x)))
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 fma(z, (1.0 - log(t)), y) + fma((a + -0.5), b, x);
}

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(fma(z, Float64(1.0 - log(t)), y) + fma(Float64(a + -0.5), b, x))
end

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[(N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original0.1
Target0.3
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

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. Simplified0.1

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error7.7
Cost8528
\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ t_2 := b \cdot \left(a + -0.5\right)\\ t_3 := t_2 + \left(y + x\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t_2 + t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{-6}:\\ \;\;\;\;\left(y + x\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 2
Error7.7
Cost8528
\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ t_2 := t_1 + \left(y + x\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\left(z + t_1\right) - z \cdot \log t\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-6}:\\ \;\;\;\;\left(y + x\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error5.1
Cost7628
\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ t_2 := \left(t_1 + \left(z + y\right)\right) - z \cdot \log t\\ \mathbf{if}\;y \leq 3.288464934908744 \cdot 10^{-73}:\\ \;\;\;\;\left(x + t_1\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;y \leq 4.1110262582815665 \cdot 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.4140216648230934 \cdot 10^{+207}:\\ \;\;\;\;t_1 + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error0.1
Cost7488
\[\left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) + \left(a \cdot b + -0.5 \cdot b\right) \]
Alternative 5
Error6.3
Cost7364
\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;y \leq 4.1110262582815665 \cdot 10^{+138}:\\ \;\;\;\;\left(x + t_1\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(y + x\right)\\ \end{array} \]
Alternative 6
Error0.1
Cost7360
\[\left(\left(z + \left(y + x\right)\right) - z \cdot \log t\right) + b \cdot \left(a + -0.5\right) \]
Alternative 7
Error6.0
Cost7240
\[\begin{array}{l} t_1 := \left(y + \left(z + x\right)\right) - z \cdot \log t\\ \mathbf{if}\;z \leq -2.0701900839372296 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.736851919285667 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error6.0
Cost7240
\[\begin{array}{l} t_1 := \left(y + x\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -2.0701900839372296 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.736851919285667 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error9.4
Cost7112
\[\begin{array}{l} t_1 := y + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -2.0701900839372296 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.736851919285667 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error9.3
Cost7112
\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -2.0701900839372296 \cdot 10^{+83}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;z \leq 9.736851919285667 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]
Alternative 11
Error10.4
Cost6984
\[\begin{array}{l} t_1 := z - z \cdot \log t\\ \mathbf{if}\;z \leq -1.0538749300575403 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.811977119018121 \cdot 10^{+161}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error10.4
Cost6984
\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.0538749300575403 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.811977119018121 \cdot 10^{+161}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error21.9
Cost968
\[\begin{array}{l} t_1 := x + b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;a + -0.5 \leq -4 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a + -0.5 \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error38.1
Cost852
\[\begin{array}{l} \mathbf{if}\;x \leq -1.66129577975365 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.953865208136259 \cdot 10^{-46}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x \leq -1.1879676747619952 \cdot 10^{-104}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.3055607845117415 \cdot 10^{-129}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x \leq -2.599009019844032 \cdot 10^{-284}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 15
Error39.1
Cost848
\[\begin{array}{l} t_1 := z + -0.5 \cdot b\\ \mathbf{if}\;y \leq -2.862072819058999 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.569320988680884 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.305884094581866 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.952330897824881 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 16
Error33.2
Cost848
\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;y \leq -1.1058912296354706 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5079165858325954 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.484375110622336 \cdot 10^{-150}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.867635379723362 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]
Alternative 17
Error19.9
Cost844
\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ t_2 := y + t_1\\ \mathbf{if}\;y \leq 3.288464934908744 \cdot 10^{-73}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;y \leq 3.952330897824881 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.087004633761351 \cdot 10^{+168}:\\ \;\;\;\;-0.5 \cdot b + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 18
Error38.3
Cost720
\[\begin{array}{l} \mathbf{if}\;y \leq -1.7539542040282362 \cdot 10^{-146}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.569320988680884 \cdot 10^{-225}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;y \leq 3.6278078239214425 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.4348466422266695 \cdot 10^{+88}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 19
Error21.4
Cost580
\[\begin{array}{l} \mathbf{if}\;y \leq 2.121904633360857 \cdot 10^{+121}:\\ \;\;\;\;x + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]
Alternative 20
Error15.6
Cost576
\[b \cdot \left(a + -0.5\right) + \left(y + x\right) \]
Alternative 21
Error30.5
Cost452
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3598259744916037 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]
Alternative 22
Error37.4
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq -3.1897615255818365 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 23
Error48.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022216 
(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)))