Average Error: 7.3 → 2.3
Time: 11.8s
Precision: binary64
Cost: 704
\[ \begin{array}{c}[y, t] = \mathsf{sort}([y, t])\\ \end{array} \]
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
↓
\[\frac{\frac{1}{z - t}}{\frac{z - y}{x}} \]
(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))
↓
(FPCore (x y z t) :precision binary64 (/ (/ 1.0 (- z t)) (/ (- z y) x)))
double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}
↓
double code(double x, double y, double z, double t) {
	return (1.0 / (z - t)) / ((z - y) / x);
}

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((y - z) * (t - z))
end function
↓
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / (z - t)) / ((z - y) / x)
end function

public static double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}
↓
public static double code(double x, double y, double z, double t) {
	return (1.0 / (z - t)) / ((z - y) / x);
}

def code(x, y, z, t):
	return x / ((y - z) * (t - z))
↓
def code(x, y, z, t):
	return (1.0 / (z - t)) / ((z - y) / x)

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end
↓
function code(x, y, z, t)
	return Float64(Float64(1.0 / Float64(z - t)) / Float64(Float64(z - y) / x))
end

function tmp = code(x, y, z, t)
	tmp = x / ((y - z) * (t - z));
end
↓
function tmp = code(x, y, z, t)
	tmp = (1.0 / (z - t)) / ((z - y) / x);
end

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[(1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\frac{\frac{1}{z - t}}{\frac{z - y}{x}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.3
Target8.0
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \]

Derivation

  1. Initial program 7.3

    \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]

  2. Simplified2.2

    \[\leadsto \color{blue}{\frac{\frac{x}{z - y}}{z - t}} \]
    Proof
    (/.f64 (/.f64 x (-.f64 z y)) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 z y))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z y))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) y)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) y))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 y z)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 y z)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x (-.f64 y z)) -1)) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 z t))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z t))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) t)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) t))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 t z)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 t z)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 t z) -1))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 x (-.f64 y z)) (*.f64 -1 (*.f64 (-.f64 t z) -1)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (-.f64 y z)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (-.f64 t z) -1) -1))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (-.f64 y z)) (Rewrite=> associate-*l*_binary64 (*.f64 (-.f64 t z) (*.f64 -1 -1)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (-.f64 y z)) (*.f64 (-.f64 t z) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 x (-.f64 y z)) (Rewrite=> *-rgt-identity_binary64 (-.f64 t z))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))): 65 points increase in error, 32 points decrease in error

  3. Applied egg-rr2.7

    \[\leadsto \color{blue}{{\left(\frac{z - y}{x} \cdot \left(z - t\right)\right)}^{-1}} \]

  4. Applied egg-rr2.3

    \[\leadsto \color{blue}{\frac{\frac{1}{z - t}}{\frac{z - y}{x}}} \]

  5. Final simplification2.3

    \[\leadsto \frac{\frac{1}{z - t}}{\frac{z - y}{x}} \]

Alternatives

Alternative 1
Error16.1
Cost1240
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{t}}{y - z}\\ \mathbf{if}\;z \leq -1.52 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.5:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error16.2
Cost1240
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{y}}{t - z}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{+27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3900:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error28.3
Cost1176
\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;t \leq -3.05 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+260}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error23.4
Cost1044
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{-x}{y}}{z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.26 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error23.4
Cost1044
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error23.2
Cost1044
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{-x}{z}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{t_2}{t}\\ \mathbf{elif}\;z \leq -3100000000:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+118}:\\ \;\;\;\;\frac{t_2}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error18.4
Cost976
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq -1500000000000:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 10^{+81}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error22.1
Cost912
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x}{z \cdot y}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error16.5
Cost844
\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{if}\;z \leq -4800:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
Alternative 10
Error16.3
Cost844
\[\begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 10^{+120}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
Alternative 11
Error4.7
Cost840
\[\begin{array}{l} t_1 := \frac{\frac{x}{z - y}}{z}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error4.7
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{z - y}{x}}\\ \end{array} \]
Alternative 13
Error11.4
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
Alternative 14
Error11.5
Cost712
\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
Alternative 15
Error11.3
Cost712
\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
Alternative 16
Error25.8
Cost584
\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -10000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 17
Error24.2
Cost584
\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 18
Error24.5
Cost584
\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 19
Error22.2
Cost584
\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 20
Error2.2
Cost576
\[\frac{\frac{x}{z - y}}{z - t} \]
Alternative 21
Error39.7
Cost320
\[\frac{x}{t \cdot y} \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

  (/ x (* (- y z) (- t z))))